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Isolated singularities of Toda equations and cyclic Higgs bundles

Qiongling Li and Takuro Mochizuki Qiongling Li:
Chern Institute of Mathematics and LPMC, Nankai University,
Tianjin 300071, China [email protected] Takuro Mochizuki:
Research Institute for Mathematical Sciences, Kyoto University,
Kyoto 606-8512, Japan [email protected]

Abstract.

This paper is the second part of our study on the Toda equations and the cyclic Higgs bundles associated with $r$ -differentials over non-compact Riemann surfaces. We classify all the solutions up to boundedness around the isolated singularity of an $r$ -differential under the assumption that the $r$ -differential is meromorphic or has some type of essential singularity. As a result, for example, we classify all the solutions on ${\mathbb{C}}$ if the $r$ -differential is a finite sum of the exponential of polynomials.

Key words and phrases:

Cyclic Higgs bundles, Toda equations, harmonic metrics, essential singularity

2010 Mathematics Subject Classification:

53C07, 58E15

1. Introduction

1.1. Harmonic bundles and Toda equations

1.1.1. Higgs bundles associated with $r$ -differentials and harmonic metrics

Let $X$ be any Riemann surface. We fix a line bundle $K_{X}^{1/2}$ with an isomorphism $K_{X}^{1/2}\otimes K_{X}^{1/2}\simeq K_{X}$ . Let $r$ be a positive integer. We set $\mathbb{K}_{X,r}:=\bigoplus_{i=1}^{r}K_{X}^{(r+1-2i)/2}$ . We define the actions of $G_{r}=\{a\in{\mathbb{C}}\,|\,a^{r}=1\}$ on $K_{X}^{(r+1-2i)/2}$ by $a\bullet v=a^{i}v$ . They induce a $G_{r}$ -action on $\mathbb{K}_{X,r}$ . For any $r$ -differential $q\in H^{0}(X,K_{X}^{r})$ , let $\theta(q)$ denote the Higgs field of $\mathbb{K}_{X,r}$ induced by the identity map $\theta(q)_{i}:K_{X}^{(r+1-2i)/2}\rightarrow K_{X}^{(r+1-2(i+1))/2}\otimes K_{X}$ $(i=1,\cdots,r-1)$ and $\theta(q)_{r}=q:K_{X}^{(1-r)/2}\rightarrow K_{X}^{(r-1)/2}\otimes K_{X}$ . Note that $\theta(q)$ is homogeneous with respect to the $G_{r}$ -action. It implies that if $h$ is a harmonic metric of $(\mathbb{K}_{X,r},\theta(q))$ then $a^{\ast}(h)$ is again a harmonic metric of $(\mathbb{K}_{X,r},\theta(q))$ .

Let $\mathop{\rm Harm}\nolimits(q)$ denote the set of $G_{r}$ -invariant harmonic metrics $h$ of $(\mathbb{K}_{X,r},\theta(q))$ such that $\det(h)=1$ . By the $G_{r}$ -invariance, the decomposition $\mathbb{K}_{X,r}=\bigoplus_{i=1}^{r}K_{X}^{(r+1-2i)/2}$ is orthogonal with respect to any $h\in\mathop{\rm Harm}\nolimits(q)$ , and hence we obtain the decomposition $h=\bigoplus h_{|K_{X}^{(r+1-2i)/2}}$ . Note that $K_{X}^{(r+1-2i)/2}$ and $K_{X}^{(r+1-2(r+1-i))/2}=K_{X}^{(-r-1+2i)/2}$ are mutually dual. We say that $h\in\mathop{\rm Harm}\nolimits(q)$ is real if $h_{|K_{X}^{(r+1-2i)/2}}$ and $h_{|K_{X}^{(-r-1+2i)/2}}$ are mutually dual. Let $\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)$ denote the subset of $h\in\mathop{\rm Harm}\nolimits(q)$ which are real.

Recall that if $X$ is compact then the classification of harmonic metrics of $(\mathbb{K}_{X,r},\theta(q))$ is well known. Indeed, if $X$ is hyperbolic, the Higgs bundle $(\mathbb{K}_{X,r}(q),\theta(q))$ is stable for any $q$ . Hence, according to the Kobayashi-Hitchin correspondence for Higgs bundles due to Hitchin and Simpson ([19], [38]), $(\mathbb{K}_{X,r},\theta(q))$ has a unique harmonic metric $h$ such that $\det(h)=1$ . Moreover, as observed by Baraglia [1], the uniqueness implies that $h$ is $G_{r}$ -invariant and real. In other words, for a compact hyperbolic Riemann surface $X$ , $\mathop{\rm Harm}\nolimits(q)=\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)$ consists of a unique element. If $X$ is an elliptic curve, it is easy to see that $\mathop{\rm Harm}\nolimits(q)$ consists of a unique element if $q\neq 0$ , and that $\mathop{\rm Harm}\nolimits(0)$ is empty. If $X$ is $\mathbb{P}^{1}$ , it is easy to see that there is no non-zero holomorphic $r$ -differential, and that $\mathop{\rm Harm}\nolimits(0)$ is empty. Note that if $X$ is compact hyperbolic and if $q=0$ , as observed by Hitchin and Simpson, $(\mathbb{K}_{X,r},\theta(0))$ with the harmonic metric is naturally a polarized variation of Hodge structure, which particularly implies the $G_{r}$ -invariance.

If $X$ is non-compact, the uniqueness of harmonic metrics of the Higgs bundle $(\mathbb{K}_{X,r},\theta(q))$ does not necessarily hold, and a harmonic metric of unit determinant is not necessarily $G_{r}$ -invariant. But, if $X$ is the complement of a finite subset in a compact Riemann surface $\overline{X}$ , and if $q$ is meromorphic on $\overline{X}$ , we may still apply the Kobayashi-Hitchin correspondence for singular Higgs bundles due to Simpson [39] in the tame case and Biquard-Boalch [4] in the wild case. (See also [29] for the extension of a wild harmonic bundle to a filtered Higgs bundle.) Indeed, in [14, 15, 30, 31], $\mathop{\rm Harm}\nolimits(q)$ was classified in the case $X={\mathbb{C}}^{\ast}$ with $q=z^{m}(dz)^{r}$ motivated by the relation with the $tt^{\ast}$ -geometry [7], and the method in [30, 31] owes on the Kobayashi-Hitchin correspondence. (See [14, 15] for a different approach to the same issue.)

In this paper, as a continuation of [24], we investigate a classification of $G_{r}$ -invariant harmonic metrics on $(\mathbb{K}_{X,r},\theta(q))$ over a more general non-compact Riemann surface $X$ with a more general $r$ -differential $q$ , which is not covered by the theory of wild harmonic bundles. In particular, we shall closely study the case where $q$ has some type of essential singularity.

1.1.2. Toda equations associated with $r$ -differentials

Let $g$ be any Kähler metric of $X$ . It induces a $G_{r}$ -invariant Hermitian metric $h^{(1)}(g)$ of $\mathbb{K}_{X,r}$ . For any other $G_{r}$ -invariant Hermitian metric $h$ such that $\det(h)=1$ , we obtain a tuple of ${\mathbb{R}}$ -valued functions ${\boldsymbol{w}}=(w_{1},\ldots,w_{r})$ such that $\sum w_{i}=0$ by the relation

h_{|K_{X}^{(r+1-2i)/2}}=e^{w_{i}}h^{(1)}(g)_{|K_{X}^{(r+1-2i)/2}}=e^{w_{i}}g^{-(r+1-2i)/2}.

Then, $h$ is contained in $\mathop{\rm Harm}\nolimits(q)$ if and only if the following type of Toda equation is satisfied:

\left\{\begin{array}[]{l}\sqrt{-1}\Lambda_{g}\partial\overline{\partial}w_{1}=e^{-w_{r}+w_{1}}|q|_{g}^{2}-e^{-w_{1}+w_{2}}-\frac{r-1}{2}k_{g}\\ \sqrt{-1}\Lambda_{g}\partial\overline{\partial}w_{i}=e^{-w_{i-1}+w_{i}}-e^{-w_{i}+w_{i+1}}-\frac{r+1-2i}{2}k_{g}\quad(i=2,\ldots,r-1)\\ \sqrt{-1}\Lambda_{g}\partial\overline{\partial}w_{r}=e^{-w_{r-1}+w_{r}}-e^{-w_{r}+w_{1}}|q|_{g}^{2}-\frac{1-r}{2}k_{g}\end{array}\right.

(1)

Here, $\Lambda_{g}$ denotes the adjoint of the multiplication of the associated Kähler form, and $k_{g}=\sqrt{-1}\Lambda_{g}R(g)$ is the Gaussian curvature of $g$ . Let $\mathop{\rm Toda}\nolimits_{1}(q,g)$ denote the set of solutions ${\boldsymbol{w}}$ of (1) satisfying $\sum w_{i}=0$ . A solution ${\boldsymbol{w}}\in\mathop{\rm Toda}\nolimits_{1}(q,g)$ is called real if $w_{i}+w_{r+1-i}=0$ . Let $\mathop{\rm Toda}\nolimits_{1}^{{\mathbb{R}}}(q,g)$ denote the set of real solutions of (1). As explained, there is a natural bijection $\mathop{\rm Harm}\nolimits(q)\simeq\mathop{\rm Toda}\nolimits_{1}(q,g)$ , which induces $\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)\simeq\mathop{\rm Toda}\nolimits_{1}^{{\mathbb{R}}}(q,g)$ .

Remark 1.1.

The equation (1) is equivalent to the Toda equation in [24]. See §1.1.5 below.

Remark 1.2.

If we change the sign in the system (1), we obtain the classical Toda equation studied extensively in integrable system, e.g. see [5, 6]. Geometrically, the classical Toda equation gives rise to harmonic maps from surface to compact flag manifolds.

Remark 1.3.

A solution of the Toda equation gives rise to an equivariant harmonic map $f$ from a universal covering of $X$ to the symmetric space $\mathop{\rm SL}\nolimits(r,{\mathbb{C}})/{\mathop{\rm SU}}(r)$ such that $\mathop{\rm Tr}\nolimits(\partial f^{\otimes i})=0$ $(i=1,\cdots,r-1)$ except for $\mathop{\rm Tr}\nolimits(\partial f^{\otimes r})$ is a nonzero constant multiple of $q$ . In lower rank, the Toda equation is encoded with much richer geometry. If $r=2$ , the Toda equation coincides with the Bochner equation for harmonic maps between surfaces for a given Hopf differential, e.g., see [37, 44, 45, 49]. For the study of solutions for given meromorphic quadratic differentials, one can check [17, 18, 50]. If $r=3$ , the Toda equation for a real solution coincides with Wang’s equation for hyperbolic affine spheres in $\mathbb{R}^{3}$ for a given Pick differential, e.g., see [3, 20, 25, 46]. For the study of solutions for given meromorphic cubic differentials, one can check [2, 10, 26, 27, 35]. If $r=4$ , the Toda equation for a real solution coincides with the Gauss-Ricci equation for maximal surfaces in $\mathbb{H}^{2,2}$ , e.g., see [8, 42].

1.1.3. Existence and uniqueness of complete solutions

A solution ${\boldsymbol{w}}\in\mathop{\rm Toda}\nolimits_{1}(q,g)$ is called complete if the metrics $e^{w_{i+1}-w_{i}}g$ $(i=1,\ldots,r-1)$ are complete. In terms of harmonic metrics, it is equivalent to the condition that the Kähler metrics $g(h)_{i}$ $(i=1,\ldots,r-1)$ induced by $h_{|K_{X}^{(r+1-2(i+1))/2}}\otimes h_{|K_{X}^{(r+1-2i)/2}}^{-1}$ are complete on $X$ , where we naturally identify the tangent bundle $K_{X}^{-1}$ of $X$ with $K_{X}^{(r+1-2(i+1))/2}\otimes(K_{X}^{(r+1-2i)/2})^{-1}$ . The following fundamental theorem is proved in [24]. (See [24] for more detailed properties of complete solutions.)

Theorem 1.4 ([24]).

Let $X$ be a non-compact Riemann surface with a holomorphic $r$ -differential $q$ . Assume either (i) $X$ is hyperbolic, or (ii) $X$ is parabolic and $q\neq 0$ . Here, we say that $X$ is hyperbolic (resp. parabolic) if a universal covering of $X$ is the upper half plane (resp. ${\mathbb{C}}$ ). Then, there uniquely exists a complete solution ${\boldsymbol{w}}^{\mathop{\rm c}\nolimits}\in\mathop{\rm Toda}\nolimits_{1}(q,g)$ . It is real, i.e., ${\boldsymbol{w}}^{\mathop{\rm c}\nolimits}\in\mathop{\rm Toda}\nolimits_{1}^{{\mathbb{R}}}(q,g)$ . Moreover, there exists a complete Kähler metric $\widetilde{g}$ of $X$ such that $e^{w_{i+1}-w_{i}}g$ $(i=1,\ldots,r-1)$ are mutually bounded with $\widetilde{g}$ , and that $|q|_{\widetilde{g}}$ is bounded. ∎

When we emphasize the dependence on $(q,g)$ , we use the notation ${\boldsymbol{w}}^{\mathop{\rm c}\nolimits}(q,g)$ . Let $h^{\mathop{\rm c}\nolimits}$ denote the corresponding harmonic metric, which is independent of the choice of $g$ . We shall also use the notation $h^{\mathop{\rm c}\nolimits}(q)$ when we emphasize the dependence on $q$ .

1.1.4. Uniqueness and non-uniqueness of general solutions

Suppose that $q$ has finitely many zeros. In this case, the results in [24] clarify whether general solutions of (1) are uniquely determined or not. Let $N$ be any relatively compact open neighbourhood of the zero set of $q$ . On $X\setminus N$ , we obtain the Kähler metric $|q|^{2/r}:=(q\cdot\overline{q})^{1/r}$ . We proved the following proposition in [24] on the uniqueness. Note that such a uniqueness was first proved in [23] for $X={\mathbb{C}}$ with a polynomial $r$ -differential $(r=2,3)$ .

Proposition 1.5 ([24]).

If $|q|^{2/r}$ induces a complete distance on $X\setminus N$ , then the equation (1) has a unique solution, i.e., $\mathop{\rm Toda}\nolimits_{1}(q,g)=\{{\boldsymbol{w}}^{\mathop{\rm c}\nolimits}(q,g)\}$ and $\mathop{\rm Harm}\nolimits(q)=\{h^{\mathop{\rm c}\nolimits}(q)\}$ . ∎

Remark 1.6.

In fact, the condition $|q|^{2/r}$ induces a complete distance on $X\setminus N$ is equivalent to that $X=\bar{X}\setminus D$ where $\bar{X}$ is compact and $D$ is finite, $q$ is meromorphic on $\bar{X}$ with poles at each point of $D$ of pole order at least $r$ . Here is a brief argument. Since $q$ has finitely many zeros, one can easily extend $|q|^{2/r}$ to a smooth metric on $X$ , which is obviously of finite total curvature. By using Huber’s theorem on complete surfaces with finite total curvature, we obtain $X=\bar{X}\setminus D$ where $\bar{X}$ is compact and $D$ is finite. The rest is proven using a similar argument as in [35, Lemma B.3].

As for the non-uniqueness, the following proposition is proved in [24].

Proposition 1.7 ([24]).

If $q$ has finitely many zeros, there exists a solution ${\boldsymbol{w}}\in\mathop{\rm Toda}\nolimits_{1}^{{\mathbb{R}}}(q,g)$ such that outside a compact subset $K$ ,

w_{i}\sim-\frac{r+1-2i}{r}\log|q|_{g},\quad 1\leq i\leq r.

In particular, solutions of (1) are not unique if $|q|^{2/r}$ does not induce a complete distance of $X\setminus N$ . ∎

When the uniqueness does not hold, it is natural to study a classification of general solutions of (1) under a mild assumption for $q$ . For example, let $X={\mathbb{C}}^{\ast}$ and $q=z^{m}(dz/z)^{r}$ $(m>0)$ . According to [14, 15, 30, 31], for any $h\in\mathop{\rm Harm}\nolimits(q)$ , we obtain a tuple of real numbers ${\boldsymbol{b}}(h)=(b(h)_{1},\ldots,b(h)_{r})$ determined by

b(h)_{i}:=\inf\Bigl{\{}c\in{\mathbb{R}}\,\Big{|}\,|z|^{c+i}|(dz)^{(r+1-2i)/2}|_{h}\,\,\mbox{ is bounded around $0$}\Bigr{\}}.

The correspondence $h\longmapsto{\boldsymbol{b}}(h)$ induces a bijection

\mathop{\rm Harm}\nolimits(q)\simeq\\ \left\{(b_{1},\ldots,b_{r})\,\left|\,b_{1}\geq b_{2}\geq\cdots\geq b_{r}\geq b_{1}-m,\,\,\sum b_{i}=-r(r+1)/2\right.\right\}.

(2)

It is our purpose in this paper to pursue a similar classification of $\mathop{\rm Harm}\nolimits(q)$ in a more general situation, which we shall explain in the following subsections.

1.1.5. Remark on the difference of the conventions for the induced Hermitian metrics

We use a different convention from [24] for the induced Hermitian metric on $K_{X}^{j/2}$ . For a Kähler metric $g=g_{0}\,dz\otimes d\overline{z}$ , we set $|(dz)^{j/2}|^{2}_{g}=(\frac{g_{0}}{2})^{-j/2}$ in this paper. (For example, see [13].) In [24], we used the norm $(|(dz)^{j/2}|^{\prime}_{g})^{2}=g_{0}^{-j/2}$ . In particular, $|q|_{g}^{2}=2^{r}(|q|^{\prime}_{g})^{2}$ . Let $h^{(0)}(g)$ denote the induced Hermitian metric of $\mathbb{K}_{X,r}$ obtained by the latter convention. Then, for an ${\mathbb{R}}^{r}$ -valued function ${\boldsymbol{u}}$ , a Hermitian metric $\bigoplus_{i=1}^{r}e^{u_{i}}h^{(0)}(g)_{|K_{X}^{(r+1-2i)/2}}$ is a harmonic metric of $(\mathbb{K}_{X,r},\theta(q))$ if and only if the following equation is satisfied, which is studied in [24]:

\left\{\begin{array}[]{l}\frac{1}{2}\sqrt{-1}\Lambda_{g}\partial\overline{\partial}u_{1}=e^{-u_{r}+u_{1}}|q|_{g}^{\prime\,2}-e^{-u_{1}+u_{2}}-\frac{r-1}{4}k_{g}\\ \frac{1}{2}\sqrt{-1}\Lambda_{g}\partial\overline{\partial}u_{i}=e^{-u_{i-1}+u_{i}}-e^{-u_{i}+u_{i+1}}-\frac{r+1-2i}{4}k_{g}\quad(i=2,\ldots,r-1)\\ \frac{1}{2}\sqrt{-1}\Lambda_{g}\partial\overline{\partial}u_{r}=e^{-u_{r-1}+u_{r}}-e^{-u_{r}+u_{1}}|q|_{g}^{\prime\,2}-\frac{1-r}{4}k_{g}\end{array}\right.

(3)

An ${\mathbb{R}}^{r}$ -valued function ${\boldsymbol{w}}$ satisfies (1) if and only if the tuple $u_{i}=w_{i}+\frac{r+1-2i}{2}\log 2$ $(i=1,\ldots,r)$ satisfies (3).

1.2. Isolated singularities

Let $D\subset X$ be a finite subset. Let us consider the case where $q$ is a holomorphic $r$ -differential on $X\setminus D$ , which is not constantly $0$ . We shall study the classification of the behaviour of $h\in\mathop{\rm Harm}\nolimits(q)$ up to boundedness around each point $P$ of $D$ . Let $(X_{P},z_{P})$ denote a holomorphic coordinate neighbourhood around $P$ with $z_{P}(P)=0$ . We set $X_{P}^{\ast}:=X_{P}\setminus\{P\}$ .

1.2.1. The case of poles

Let us recall that if $P$ is a pole of $q$ , a general theory of harmonic bundles [38, 29, 32] allows us to obtain the classification in terms of parabolic structure up to boundedness. We have the expression

q_{|X_{P}^{\ast}}=z_{P}^{m_{P}}\alpha_{P}\cdot(dz_{P}/z_{P})^{r},

where $m_{P}$ denotes an integer, and $\alpha_{P}$ induces a holomorphic function on $X_{P}$ such that $\alpha_{P}(P)\neq 0$ . If $X_{P}$ is sufficiently small, $\alpha_{P}$ is nowhere vanishing.

If $m_{P}\leq 0$ , we obtain the following estimate:

\log\bigl{|}(dz_{P})^{(r+1-2i)/2}\bigr{|}_{h}+\frac{r+1-2i}{2r}\log|z_{P}|^{m_{P}-r}=O(1).

In particular, any $h_{1}$ and $h_{2}\in\mathop{\rm Harm}\nolimits(q)$ are mutually bounded around $P$ . (See §3.5.1 for more details.)

If $m_{P}>0$ , there exists ${\boldsymbol{b}}_{P}(h)=(b_{P,1}(h),\ldots,b_{P,r}(h))\in{\mathbb{R}}^{r}$ satisfying

\left\{\begin{array}[]{l}b_{P,1}(h)\geq b_{P,2}(h)\geq\cdots\geq b_{P,r}(h)\geq b_{P,1}(h)-m_{P},\\ \sum_{i=1}^{r}b_{P,i}(h)=-\frac{r(r+1)}{2},\end{array}\right.

(4)

for which the following estimates hold:

\log\bigl{|}(dz_{P})^{(r+1-2i)/2}\bigr{|}_{h}+(b_{P,i}(h)+i)\log|z_{P}|-\frac{k_{i}({\boldsymbol{b}}_{P}(h))}{2}\log\bigl{(}-\log|z_{P}|\bigr{)}\\ =O(1).

(5)

Here, ${\boldsymbol{k}}({\boldsymbol{b}}_{P}(h))=(k_{1}({\boldsymbol{b}}_{P}(h)),\ldots,k_{r}({\boldsymbol{b}}_{P}(h)))$ denotes the tuple of integers determined by ${\boldsymbol{b}}_{P}(h)$ as in §3.5.2. It particularly implies that $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ are mutually bounded around $P$ if and only if ${\boldsymbol{b}}_{P}(h_{1})={\boldsymbol{b}}_{P}(h_{2})$ . (See §3.5.2 for more details.)

1.2.2. Holomorphic functions with multiple growth orders

To study more general cases, we introduce a reasonable subclass of essential singularities of holomorphic functions. Let $f$ denote a holomorphic function on $X_{P}^{\ast}$ .

Let $\varpi:\widetilde{X}_{D}\longrightarrow X$ denote the oriented real blowing up. Let $Q\in\varpi^{-1}(P)$ . We fix a branch of $\log z_{P}$ around $Q$ .

•

We say that $f$ is regularly bounded around $Q$ if there exist a neighbourhood $\mathcal{U}$ of $Q$ in $\widetilde{X}_{D}$ , non-zero complex numbers $\alpha_{j}$ $(j=1,\ldots,m)$ , and mutually distinct real numbers $c_{j}$ $(j=1,\ldots,m)$ for which $\bigl{|}f-\sum_{j=1}^{m}\alpha_{j}z_{P}^{\sqrt{-1}c_{j}}\bigr{|}\to 0$ holds as $|z_{P}|\to 0$ in $\mathcal{U}\setminus\varpi^{-1}(P)$ .
•

We say that $f$ has a single growth order at $Q$ if there exists $\mathfrak{a}(f,Q)\in\bigoplus_{b>0}{\mathbb{C}}z_{P}^{-b}$ , $a(f,Q)\in{\mathbb{R}}$ and $j(f,Q)\in{\mathbb{Z}}_{\geq 0}$ such that $e^{-\mathfrak{a}(f,Q)}z_{P}^{-a(f,Q)}(\log z_{P})^{-j(f,Q)}f$ is regularly bounded around $Q$ .
•

We say that $f$ has multiple growth orders at $Q$ if $f$ is expressed as a finite sum of holomorphic functions $f_{i}$ $(i=1,\ldots,m)$ with a single growth order at $Q$ such that $\mathfrak{a}(f_{i},Q)$ $(i=1,\ldots,m)$ are mutually distinct.
•

We say that $f$ has multiple growth orders at $P$ if $f$ has multiple growth orders at any $Q\in\varpi^{-1}(P)$ .

Remark 1.8.

If $f$ satisfies a linear differential equation

\partial_{z_{P}}^{n}f+\sum_{j=0}^{n-1}\gamma_{j}(z_{P})\partial_{z_{P}}^{j}f=0

for meromorphic functions $\gamma_{j}$ on $(X_{P},P)$ , then $f$ has multiple growth orders at $P$ . (For example, see [28, §II.1].)

Note that when $f$ has single growth order at $Q\in\varpi^{-1}(P)$ ,

|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))

induces a continuous function on a neighbourhood of $Q$ . The function is also denoted by $|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))$ . As the value of the function at $Q$ , we obtain a real number $|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))(Q)$ .

If $f$ has multiple growth orders at $P$ , there exists a finite subset $\mathcal{Z}(f)\subset\varpi^{-1}(P)$ such that the following holds at $Q\in\varpi^{-1}(P)\setminus\mathcal{Z}(f)$ .

•

$f$ has a single growth order at $Q$ .
•

$|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))(Q)\neq 0$ ,

Note that $\varpi^{-1}(P)$ $(P\in D)$ are identified with $S^{1}$ . An interval $I$ of $\varpi^{-1}(P)$ is called special with respect to $f$ if there exists $\alpha_{I}\in{\mathbb{C}}^{\ast}$ and $\rho_{I}>0$ such that the following holds.

•

The length of $I$ is $\pi/\rho_{I}$ .
•

For any $Q\in I\setminus\mathcal{Z}(f)$ , we obtain that

$\mathfrak{a}(f,Q)-\alpha_{I}z_{P}^{-\rho_{I}}\in\bigoplus_{0<b<\rho_{I}}{\mathbb{C}}z_{P}^{-b},$

and that $|\alpha_{I}z_{P}^{-\rho_{I}}|^{-1}\mathop{\rm Re}\nolimits(\alpha_{I}z_{P}^{-\rho_{I}})(Q)<0$ . Here, we assume that the branches of $\log z_{P}$ at $Q$ are analytically continued along $I$ .

1.2.3. The case where $q$ is not meromorphic but has multiple growth orders

For $P\in D$ , we have the expression $q=f_{P}(dz_{P})^{r}$ , where $f_{P}$ is a holomorphic function on $X_{P}^{\ast}$ . We say that $q$ has multiple growth orders at $P$ , if $f_{P}$ has multiple growth orders at $P$ .

Assume that $q$ is not meromorphic but has multiple growth orders at $P$ . Let $\mathcal{S}(q,P)$ denote the (possibly empty) set of intervals in $\varpi^{-1}(P)$ which are special with respect to $f_{P}$ . Let $\mathcal{P}$ denote the set of ${\boldsymbol{a}}\in{\mathbb{R}}^{r}$ such that

a_{1}\geq a_{2}\geq\cdots\geq a_{r}\geq a_{1}-1,\quad\sum a_{i}=0.

Theorem 1.9 (Theorem 6.1).

•

For any $h\in\mathop{\rm Harm}\nolimits(q)$ , there exist ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ $(I\in\mathcal{S}(q,P))$ and $\epsilon>0$ such that the following estimates hold as $|z_{P}|\to 0$ on $\bigl{\{}|\arg(\alpha_{I}z_{P}^{-\rho(I)})-\pi|<(1-\delta)\pi/2\bigr{\}}$ for any $\delta>0$ :

\log\bigl{|}(dz_{P})^{(r+1-2i)/2}\bigr{|}_{h}+a_{i}(h)\mathop{\rm Re}\nolimits\bigl{(}\alpha_{I}z_{P}^{-\rho_{I}}\bigr{)}=O\bigl{(}|z_{P}|^{-\rho_{I}+\epsilon}\bigr{)}.

(6)

•

If ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ for any $I\in\mathcal{S}(q,P)$ , there exists a relatively compact neighbourhood $X_{P}$ of $P$ such that $h_{1}$ and $h_{2}$ are mutually bounded on $X_{P}^{\ast}$ .
•

For any ${\boldsymbol{a}}_{I}\in\mathcal{P}$ $(I\in\mathcal{S}(q,P))$ , there exists $h\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}_{I}$ $(I\in\mathcal{S}(q,P))$ .

Let us explain rough ideas for the proof. Let $h\in\mathop{\rm Harm}\nolimits(q)$ . If $f_{P}$ has a single growth order at $Q$ with

|\mathfrak{a}(f_{P},Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f_{P},Q))(Q)>0,

then $h$ should be close to a canonical harmonic metric $h_{\mathop{\rm can}\nolimits}$ around $Q$ outside of some neighbourhoods of the zeroes of $q$ , which follows from variants of Simpson’s main estimate. (See Proposition 5.1. See §3.2.1 for $h_{\mathop{\rm can}\nolimits}$ .) Therefore, around such $Q$ , we may apply the subharmonicity of the difference of two harmonic metrics (see §2.4) to obtain that two harmonic metrics are mutually bounded. Let $I$ be an interval of $\varpi^{-1}(P)$ such that

|\mathfrak{a}(f_{P},Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f_{P},Q))(Q)<0

for any $Q\in I\setminus\mathcal{Z}(f_{P})$ . Assume that $I$ is maximal among such intervals. There exist $\rho_{I}\in{\mathbb{R}}_{>0}$ and $\alpha_{I}\in{\mathbb{C}}^{\ast}$ such that $\mathfrak{a}(f_{P},Q)-\alpha_{I}z_{P}^{-\rho_{I}}\in\bigoplus_{0<b<\rho_{I}}{\mathbb{C}}z_{P}^{-b}$ for any $Q\in I\setminus\mathcal{Z}(f)$ . Moreover, the length of $I$ is not strictly greater than $\pi/\rho_{I}$ . If the length of $I$ is strictly smaller than $\pi/\rho_{I}$ , we can apply Phragmén-Lindelöf theorem (Proposition 4.1) to obtain that any $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ are mutually bounded around the closure $\overline{I}$ of $I$ (Theorem 5.2). If the length of $I$ is $\pi/\rho$ , i.e., if $I$ is special, then the Nevanlinna formula (Proposition 4.4) says that there exists the tuple ${\boldsymbol{a}}_{I}(h)$ for any $h\in\mathop{\rm Harm}\nolimits(q)$ such that the estimate (6) holds. (See Theorem 5.3.) We can also apply Phragmén-Lindelöf theorem to obtain that any $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ are mutually bounded around $\overline{I}$ if ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ .

For the construction of $h\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}_{I}$ $(I\in\mathcal{S}(q,P))$ , we shall find a neighbourhood $\mathcal{U}_{I}$ of $I\in\mathcal{S}(q,P)$ , a $G_{r}$ -invariant Hermitian metric $h_{{\boldsymbol{a}}}$ of $\mathbb{K}_{X\setminus D,r}$ with $\det(h_{{\boldsymbol{a}}})=1$ , and an increasing sequence $Y_{i}$ of relatively compact open subsets of $X\setminus D$ such that the following holds.

•

The estimate (6) holds for $h_{{\boldsymbol{a}}}$ and each $I\in\mathcal{S}(q,P)$ .
•

The boundary $\partial Y_{i}$ are smooth, and $\bigcup Y_{i}=X\setminus D$ .
•

For $h_{i}\in\mathop{\rm Harm}\nolimits(q_{|Y_{i}})$ such that $h_{i|\partial Y_{i}}=h_{{\boldsymbol{a}}|\partial Y_{i}}$ , we obtain $\log\mathop{\rm Tr}\nolimits(h_{i}\cdot h_{{\boldsymbol{a}}|Y_{i}}^{-1})\leq C|z_{P}|^{-\rho_{I}+\epsilon}$ on $Y_{i}\cap(\mathcal{U}_{I}\setminus\varpi^{-1}(P))$ , where $C>0$ and $0<\epsilon<\rho_{I}$ are independent of $i$ .

Note that there always exists $h_{i}\in\mathop{\rm Harm}\nolimits(q_{|Y_{i}})$ such that $h_{i|\partial Y_{i}}=h_{{\boldsymbol{a}}|\partial Y_{i}}$ by a theorem of Donaldson [9] (see Proposition 2.1). For any compact subset $K\subset X\setminus D$ , there exists $C_{K}>0$ such that $\mathop{\rm Tr}\nolimits(h_{i}\cdot h_{{\boldsymbol{a}}|Y_{i}}^{-1})<C_{K}$ on $K$ for any $i$ such that $K\subset Y_{i}$ . By taking a subsequence, we may assume that the sequence $h_{i}$ is convergent to $h_{\infty}\in\mathop{\rm Harm}\nolimits(q)$ . (See Proposition 3.15.) By the uniform estimate $\log\mathop{\rm Tr}\nolimits(h_{i}h_{{\boldsymbol{a}}|Y_{i}}^{-1})\leq C|z_{P}|^{-\rho_{I}+\epsilon}$ on $Y_{i}\cap(\mathcal{U}_{I}\setminus\varpi^{-1}(P))$ , we obtain the estimate $\log\mathop{\rm Tr}\nolimits(h_{\infty}h_{{\boldsymbol{a}}}^{-1})\leq C|z_{P}|^{-\rho_{I}+\epsilon}$ on $\mathcal{U}_{I}\setminus\varpi^{-1}(P)$ , and hence ${\boldsymbol{a}}_{I}(h_{\infty})={\boldsymbol{a}}_{I}$ .

1.3. Global classification in the case where $q$ has multiple growth orders

Suppose that $q$ has multiple growth orders at each point of $D$ . Let $D_{\mathop{\rm mero}\nolimits}$ denote the set of $P\in D$ such that $q$ is meromorphic at $P$ . We divide $D_{\mathop{\rm mero}\nolimits}=D_{>0}\sqcup D_{\leq 0}$ , where $D_{>0}$ denotes the set of $P\in D$ such that $m_{P}>0$ , and $D_{\leq 0}:=D\setminus D_{>0}$ . For $P\in D_{>0}$ , let $\mathcal{P}(q,P)$ denote the set of ${\boldsymbol{b}}\in{\mathbb{R}}^{r}$ satisfying the condition (4). We set $D_{\mathop{\rm ess}\nolimits}:=D\setminus D_{\mathop{\rm mero}\nolimits}$ . We set $\mathcal{S}(q):=\coprod_{P\in D_{\mathop{\rm ess}\nolimits}}\mathcal{S}(q,P)$ . Then, we obtain the map

\mathop{\rm Harm}\nolimits(q)\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}(q,P)\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}.

(7)

If $D_{>0}$ is empty, $\prod_{P\in D_{>0}}\mathcal{P}(q,P)$ denotes a set which consists of one element. Similarly, if $\mathcal{S}(q)$ is empty, $\prod_{I\in\mathcal{S}(q)}\mathcal{P}$ denotes a set which consists of one element.

Let $\mathcal{P}^{{\mathbb{R}}}(q,P)$ denote the set of ${\boldsymbol{b}}\in\mathcal{P}(q,P)$ such that $b_{i}+b_{r+1-i}=-r-1$ . Let $\mathcal{P}^{{\mathbb{R}}}\subset\mathcal{P}$ denote the set of ${\boldsymbol{a}}\in\mathcal{P}$ such that $a_{i}+a_{r+1-i}=0$ . The map (7) induces the following:

\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}^{{\mathbb{R}}}(q,P)\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}^{{\mathbb{R}}}.

(8)

The following theorem is a special case of Theorem 1.12 below.

Theorem 1.10.

If $X$ is compact, then the maps (7) and (8) are bijective.

As a special case of Theorem 1.10, we obtain the following corollary in the meromorphic case.

Corollary 1.11.

Suppose that $X$ is compact, and that $q$ is meromorphic on $(X,D)$ . Then, there exists a natural bijection $\mathop{\rm Harm}\nolimits(q)\simeq\prod_{P\in D_{>0}}\mathcal{P}(q,P)$ . If moreover $m_{P}\leq 0$ at each $P\in D$ , we obtain $\mathop{\rm Harm}\nolimits(q)=\{h^{\mathop{\rm c}\nolimits}(q)\}$ . ∎

If $X$ is not compact, it is reasonable to impose an additional boundary condition on the behaviour of harmonic metrics around the infinity of $X$ . Recall that $D$ is assumed to be finite. Let $\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ denote the set of $h\in\mathop{\rm Harm}\nolimits(q)$ such that for any relatively compact open neighbourhood $N$ of $D$ , the Kähler metrics $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are complete. (See §1.1.3 for $g(h)_{i}$ .) It turns out that $h\in\mathop{\rm Harm}\nolimits(q)$ is contained in $\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ if and only if for any relatively compact open neighbourhood $N$ of $D$ , $h_{|X\setminus N}$ and $h^{\mathop{\rm c}\nolimits}(q)_{|X\setminus N}$ are mutually bounded. We set $\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q;D,\mathop{\rm c}\nolimits):=\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)\cap\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ .

We obtain the following map as the restriction of (7):

\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}(q,P)\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}.

(9)

Similarly, we obtain the following map as the restriction of (8):

\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q;D,\mathop{\rm c}\nolimits)\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}^{{\mathbb{R}}}(q,P)\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}^{{\mathbb{R}}}.

(10)

Theorem 1.12 (Theorem 6.6).

The maps (9) and (10) are bijective.

Let us explain rough ideas for the proof of Theorem 1.12. For the uniqueness, the argument in [24] is available even in this situation. According to [38, Lemma 3.1] (see §2.4), for any two $h_{j}\in\mathop{\rm Harm}\nolimits(q)$ $(j=1,2)$ , we obtain the inequality (128) on $X\setminus D$ . If ${\boldsymbol{b}}_{P}(h_{1})={\boldsymbol{b}}_{P}(h_{2})$ $(P\in D_{>0})$ and ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ $(I\in\mathcal{S}(q))$ , we obtain that $h_{1}$ and $h_{2}$ are mutually bounded, and hence [39, Lemma 2.2] implies that the inequality (128) weakly holds on $X$ . Then, as in [24], we may apply Omori-Yau maximum principle to obtain $h_{1}=h_{2}$ . Note that the argument can be simplified if $X$ is compact. Indeed, because $X\setminus D$ is potential theoretically parabolic in this case, we obtain $h_{1}=h_{2}$ immediately from the mutual boundedness of $h_{1}$ and $h_{2}$ . (See Corollary 8.19 and Remark 8.20.)

As for the existence, for any $({\boldsymbol{b}}_{P})_{P\in D_{>0}}$ and $({\boldsymbol{a}}_{I})_{I\in\mathcal{S}(q)}$ , we can construct a $G_{r}$ -invariant Hermitian metric $h_{0}$ of $\mathbb{K}_{X\setminus D,r}$ such that the following holds.

•

There exists a relatively compact neighbourhood $N_{1}$ of $D$ such that $h_{0|X\setminus N_{1}}=h^{\mathop{\rm c}\nolimits}(q)_{|X\setminus N_{1}}$ .

•

There exists a relatively compact neighbourhood $N_{2}\subset N_{1}$ such that $h_{0|N_{2}\setminus D}\in\mathop{\rm Harm}\nolimits(q_{|N_{2}\setminus D})$ and that

{\boldsymbol{b}}_{P}(h_{0|N_{2}\setminus D})={\boldsymbol{b}}_{P}\quad(P\in D_{>0}),\quad\quad{\boldsymbol{a}}_{I}(h_{0|N_{2}\setminus D})={\boldsymbol{a}}_{I}\quad(I\in\mathcal{S}(q)).

Note that there exists a compact subset $K\subset X\setminus D$ such that $h_{0|X\setminus K}\in\mathop{\rm Harm}\nolimits(q_{|X\setminus K})$ . We develop a variant of Kobayashi-Hitchin correspondence for harmonic bundles (Proposition 2.11, Proposition 3.18), which allows us to obtain $h\in\mathop{\rm Harm}\nolimits(q)$ such that $h$ and $h_{0}$ are mutually bounded. We remark that if $X$ is compact, the argument can be simplified, again. Indeed, we immediately obtain the desired metric by applying a theorem of Simpson [38] to $h_{0}$ .

1.4. Comparison of the methods and the results

In [24] and this paper, we apply rather different two methods. The both methods have their own advantage. In fact, one of the goals of this work is to present readers as many as tools which can be useful in dealing with cyclic Higgs bundles over non-compact Riemann surfaces.

On one hand, the study in [24] is more p.d.e theoretic, and it is available in a quite general setting. The p.d.e tools like Omori-Yau and Cheng-Yau maximum principles work well for complete Riemann manifolds with bounded curvature. The method is particularly powerful in analyzing real solutions of (1), which allows us to obtain precise estimates of complete solutions. Therefore, we manage to use the maximum principles together with the method of super-subsolution to construct a complete solution for any Riemann surface and any holomorphic $r$ -differential.

On the other hand, the study in this paper heavily owes to the techniques and tools developed in the theory of Kobayashi-Hitchin correspondence for Higgs bundles, pioneered by Donaldson, Hitchin and Simpson. It is more efficient if the global information on $q$ is provided. In the case of $(\mathbb{K}_{X,r},\theta(q))$ , the Higgs field is generically regular semisimple, and the spectral curve is easily described as the set of $r$ -th roots of $q$ . It allows us to use efficiently variants of Simpson’s main estimate (see §3.3) in controlling the behaviour of harmonic metrics on $(\mathbb{K}_{X,r},\theta(q))$ around the isolated singularities by using the information of $q$ . Therefore, for instance, under the assumptions that $X$ is compact, that $D$ is finite, and that $q$ has at most multiple growth orders at each point of $D$ , we can classify the set of solutions of the Toda equation associated with $q$ .

In the case that both methods can be applied, we obtain different features of solutions from these two methods. Suppose that $X$ is compact, and that $q$ has multiple growth orders at each point of $D$ . Using Theorem 1.10, we obtain the full set $\mathop{\rm Harm}\nolimits(q)$ of solutions $h^{\mathcal{B},\mathcal{A}}$ , where $(\mathcal{B},\mathcal{A})=\big{(}({\boldsymbol{b}}_{P})_{P\in D_{>0}},({\boldsymbol{a}}_{I})_{I\in\mathcal{S}(q)}\big{)}\in\prod_{P\in D_{>0}}\mathcal{P}(q,P)\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}$ satisfies

	$\displaystyle\sum_{i}b_{P,i}=-\frac{r(r+1)}{2},\quad b_{P,i}\geq b_{P,i+1}\quad(i=1,\cdots,r-1),$
	$\displaystyle b_{P,r}\geq b_{P,1}-m_{P},\quad\text{for $P\in D_{>0}$},$
	$\displaystyle\sum_{i}a_{I,i}=0,\quad a_{I,1}\geq a_{I,2}\geq\cdots\geq a_{I,r}\geq a_{I,1}-1,\quad\text{for $I\in\mathcal{S}(q)$}.$

Suppose in addition that $q$ has finitely many zeros. ¿From Theorem 1.4 and Proposition 1.7, two special solutions are singled out. One is the complete solution in Theorem 1.4, which corresponds to

	$\displaystyle b_{P,i}=-\frac{r+1}{2},\quad\text{for $P\in D_{>0}$},$
	$\displaystyle a_{I,i}=0,\quad\text{for $I\in\mathcal{S}(q)$}.$

The other one is the real solution purely controlled by $q$ which corresponds to

	$\displaystyle b_{P,i}=\frac{r+1-2i}{2r}m_{P}-\frac{r+1}{2},\quad\text{for $P\in D_{>0}$},$
	$\displaystyle a_{I,i}=\frac{r+1-2i}{2r},\quad\text{for $I\in\mathcal{S}(q)$}.$

1.5. A generalization

Assume that $X\setminus D$ is non-compact, i.e., $X$ is non-compact, or $X$ is compact and $D$ is non-empty. (See Remark 7.12 for the case that $X\setminus D$ is compact, i.e., $X$ is compact and $D$ is empty.) Let $L_{i}$ $(i=1,\ldots,r)$ be holomorphic line bundles on $X\setminus D$ . Let $\psi_{i}:L_{i}\longrightarrow L_{i+1}\otimes K_{X\setminus D}$ $(i=1,\cdots,r-1)$ and $\psi_{r}:L_{r}\longrightarrow L_{1}\otimes K_{X\setminus D}$ be non-zero morphisms. We set $E:=\bigoplus L_{i}$ . Let $\theta$ be the cyclic Higgs field of $E$ induced by $\psi_{i}$ $(i=1,\ldots,r)$ .

We obtain a holomorphic section $q:=\psi_{r}\circ\cdots\circ\psi_{1}$ of $K_{X\setminus D}^{\otimes r}$ , and a holomorphic section $q_{\leq r-1}:=\psi_{r-1}\circ\cdots\circ\psi_{1}$ of $\mathop{\rm Hom}\nolimits(L_{1},L_{r})\otimes K_{X\setminus D}^{\otimes(r-1)}$ . We assume the following.

•

The zero set of $q_{\leq r-1}$ is finite.
•

$q$ is not constantly $0$ . Moreover, $q$ has multiple growth orders at each $P\in D$ .

Let $D_{\mathop{\rm mero}\nolimits}$ denote the set of $P\in D$ such that $q$ is meromorphic at $P$ . We put $D_{\mathop{\rm ess}\nolimits}:=D\setminus D_{\mathop{\rm mero}\nolimits}$ . For $P\in D_{\mathop{\rm mero}\nolimits}$ , we describe $q_{|X_{P}^{\ast}}=z_{P}^{m_{P}}\beta_{P}(dz_{P}/z_{P})^{r}$ , where $\beta_{P}$ is a nowhere vanishing holomorphic function on $X_{P}$ . Let $D_{>0}$ denote the set of $P\in D_{\mathop{\rm mero}\nolimits}$ such that $m_{P}>0$ . We set $D_{\leq 0}:=D_{\mathop{\rm mero}\nolimits}\setminus D_{>0}$ .

Let $h_{\det(E)}$ be a flat metric of $\det(E)$ . For each $P\in D$ , there exist a tuple of frames ${\boldsymbol{v}}_{P}=(v_{P,i}\,|\,i=1,\ldots,r)$ of $L_{i|X_{P}^{\ast}}$ and a real number $c({\boldsymbol{v}}_{P})$ such that $\theta(v_{P,i})=v_{P,i+1}dz_{P}/z_{P}$ $(i=1,\ldots,r-1)$ and

\bigl{|}v_{P,1}\wedge\cdots\wedge v_{P,r}\bigr{|}_{h_{\det(E)}}=|z_{P}|^{-c({\boldsymbol{v}}_{P})}.

Let $\mathcal{P}(q,P,{\boldsymbol{v}}_{P})$ be the set of ${\boldsymbol{b}}=(b_{i})\in{\mathbb{R}}^{r}$ such that

\sum_{i=1}^{r}b_{i}=c({\boldsymbol{v}}_{P}),\quad b_{i}\geq b_{i+1}\,\,(i=1,\ldots,r-1),\quad b_{r}\geq b_{1}-m_{P}.

Remark 1.13.

If $(E,\theta)=(\mathbb{K}_{X\setminus D,r},\theta(q))$ , we canonically choose

v_{P,i}=z_{P}^{i}(dz_{P})^{(r+1-2i)/2}.

Because $\det(\mathbb{K}_{X\setminus D,r})=\mathcal{O}_{X\setminus D}$ , we choose $h_{\det(E)}=1$ . Then, $c({\boldsymbol{v}}_{P})=-r(r+1)/2$ .

We consider the $G_{r}$ -action on $L_{i}$ by $a\bullet u_{i}=a^{i}u_{i}$ , which induces a $G_{r}$ -action on $E$ . Let $\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ denote the set of $G_{r}$ -invariant harmonic metrics $h$ of $(E,\theta)$ such that $\det(h)=h_{\det(E)}$ . We recall that $D$ is assumed to be a finite subset of $X$ .

Proposition 1.14 (Proposition 7.5, Proposition 7.6).

The following holds for any $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ .

•

For any $P\in D_{>0}$ , there exists ${\boldsymbol{b}}_{P}(h)\in\mathcal{P}(q,P,{\boldsymbol{v}}_{P})$ determined by the following condition.

$b_{P,i}(h)=\inf\Bigl{\{}b\in{\mathbb{R}}\,\Big{|}\,\mbox{\rm$|z_{P}|^{b}|v_{P,i}|_{h}$ is bounded}\Bigr{\}}.$

•

For any $P\in D_{\mathop{\rm ess}\nolimits}$ and any $I\in\mathcal{S}(q,P)$ , there exist ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ and $\epsilon>0$ such that the following estimates hold as $|z_{P}|\to 0$ on $\bigl{\{}|\arg(\alpha_{I}z_{P}^{-\rho(I)})-\pi|<(1-\delta)\pi/2\bigr{\}}$ for any $\delta>0$ :

\log|v_{i}|_{h}+a_{I,i}(h)\mathop{\rm Re}\nolimits(\alpha_{I}z_{P}^{-\rho(I)})=O\bigl{(}|z_{P}|^{-\rho(I)+\epsilon}\bigr{)}

(See §1.2.2 for $\alpha_{I}$ and $\rho(I)$ for a special interval $I$ .)

Moreover, if $h_{i}\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ satisfy ${\boldsymbol{b}}_{P}(h_{1})={\boldsymbol{b}}_{P}(h_{2})$ $(P\in D_{>0})$ and ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ $(I\in\coprod_{P\in D_{\mathop{\rm ess}\nolimits}}\mathcal{S}(q,P))$ , then $h_{1}$ and $h_{2}$ are mutually bounded on $N\setminus D$ for any relatively compact neighbourhood $N$ of $D$ .

Let $Z(q_{\leq r-1})$ denote the zero set of $q_{\leq r-1}$ , which is assumed to be finite. We set $\widetilde{D}=D\cup Z(q_{\leq r-1})$ . Note that $\psi_{i}$ $(i=1,\ldots,r-1)$ induce isomorphisms $K_{X\setminus\widetilde{D}}^{-1}\simeq(L_{i+1}/L_{i})_{|X\setminus\widetilde{D}}$ . Hence, for any $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ , we obtain Kähler metrics $g(h)_{i}$ $(i=1,\ldots,r-1)$ induced by the restrictions of $h$ to $L_{i|X\setminus\widetilde{D}}$ and $L_{i+1|X\setminus\widetilde{D}}$ . Let

\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)

denote the set of $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ such that for any relatively compact open neighbourhood $N$ of $\widetilde{D}$ , the metrics $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are complete.

By Proposition 1.14, we obtain the map

\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}(q,P,{\boldsymbol{v}}_{P})\times\prod_{P\in D_{\mathop{\rm ess}\nolimits}}\prod_{I\in\mathcal{S}(q,P)}\mathcal{P}.

(11)

Theorem 1.15 (Theorem 7.11).

The map (11) is a bijection.

1.6. Examples on ${\mathbb{C}}$

Let $\gamma(z)$ be any non-zero polynomial. We set $q=\gamma(z)e^{\sqrt{-1}z}(dz)^{r}$ . It is easy to see that $\{\infty\}\times\{0<\arg(z)<\pi\}$ is the unique special interval with respect to $q$ in this case. Hence, we obtain the following proposition as a corollary of Theorem 1.9.

Proposition 1.16 (A special case of Proposition 6.12).

We have the bijection $\mathop{\rm Harm}\nolimits(q)\simeq\mathcal{P}$ . ∎

More generally, let $\gamma_{i}(z)$ $(i=1,\ldots,m)$ be non-zero polynomials. Let $\alpha_{i}$ $(i=1,\ldots,m)$ be mutually distinct complex numbers. We set $q=\sum_{i=1}^{m}\gamma_{i}(z)e^{\sqrt{-1}\alpha_{i}z}(dz)^{r}$ .

Proposition 1.17 (A special case of Proposition 6.14).

If there exists a non-zero complex number $\alpha$ such that $\alpha_{i}/\alpha\in{\mathbb{R}}_{>0}$ $(i=1,\ldots,m)$ , then there exists a bijection $\mathop{\rm Harm}\nolimits(q)\simeq\mathcal{P}$ . Otherwise, $\mathop{\rm Harm}\nolimits(q)=\{h^{\mathop{\rm c}\nolimits}\}$ . ∎

Corollary 1.18.

For any non-zero polynomial $\gamma(z)$ , we obtain that $\mathop{\rm Harm}\nolimits\bigl{(}\gamma(z)\cos(z)(dz)^{r}\bigr{)}=\{h^{\mathop{\rm c}\nolimits}\}$ . ∎

We may apply Theorem 1.10 to the case $q=f(dz)^{r}$ in which $f$ is the product of a non-zero polynomial and a non-zero solution of a linear differential equation with polynomial coefficients. For instance, let $\mathop{\rm Ai}\nolimits(z)$ be the Airy function given as in (129) below, which is a solution of the differential equation $\partial_{z}^{2}u-zu=0$ . Let $\gamma(z)$ be any non-zero polynomial. For the $r$ -differential $q=\gamma(z)\mathop{\rm Ai}\nolimits(z)(dz)^{r}$ , $\{-\pi/3<\arg(z)<\pi/3\}$ is the unique special interval in this case. (See (130). See [47, §23] for a more detailed asymptotic expansion of $\mathop{\rm Ai}\nolimits(z)$ .) Hence, we obtain the following.

Proposition 1.19 (Proposition 6.16).

If $q=\gamma(z)\mathop{\rm Ai}\nolimits(z)(dz)^{r}$ , we have a bijection $\mathop{\rm Harm}\nolimits(q)\simeq\mathcal{P}$ . ∎

1.7. Acknowledgement

This study starts from a discussion during the workshop “Higgs bundles and related topics” in University of Nice, in 2017. The first author is partially supported by the National Key R $\&$ D Program of China No. 2022YFA1006600, the Fundamental Research Funds for the Central Universities and Nankai Zhide foundation. The second author is grateful to Martin Guest and Claus Hertling for discussions on harmonic bundles and Toda equations. The second author is partially supported by the Grant-in-Aid for Scientific Research (S) (No. 16H06335), the Grant-in-Aid for Scientific Research (S) (No. 17H06127), the Grant-in-Aid for Scientific Research (A) (No. 21H04429), the Grant-in-Aid for Scientific Research (A) (No. 22H00094), the Grant-in-Aid for Scientific Research (A) (No. 23H00083), the Grant-in-Aid for Scientific Research (C) (No. 15K04843), and the Grant-in-Aid for Scientific Research (C) (No. 20K03609), Japan Society for the Promotion of Science. The second author is also partially supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

2. Some existence results of harmonic metrics

2.1. Dirichlet problem

Let $X$ be any Riemann surface. Let $(E,\overline{\partial}_{E},\theta)$ be a Higgs bundle on $X$ . Let $Y\subset X$ be a relatively compact connected open subset with smooth boundary $\partial Y$ . Assume that $\partial Y$ is non-empty. Let $h_{\partial Y}$ be any Hermitian metric of $E_{|\partial Y}$ . The following proposition is essentially due to Donaldson [9].

Proposition 2.1 (Donaldson).

There exists a unique harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)$ such that $h_{|\partial Y}=h_{\partial Y}$ .

Proof Donaldson proved this theorem in the case where $Y$ is a disc in [9]. The general case is similar. We shall give a proof for the convenience of the readers. We may assume that $X$ is an open Riemann surface. According to [16], there exists a nowhere vanishing holomorphic $1$ -form $\tau$ om $X$ . Let $f$ be the automorphism of $E$ determined by $\theta=f\,\tau$ . We consider the Kähler metric $g_{X}=\tau\,\overline{\tau}$ of $X$ .

Let $\Gamma$ be a lattice of ${\mathbb{C}}$ and let $T$ be a real $2$ -dimensional torus obtained as ${\mathbb{C}}/\Gamma$ . We set $g_{T}=dz\,d\overline{z}$ . We set $\widetilde{X}:=X\times T$ with the projection $p:\widetilde{X}\longrightarrow X$ . It is equipped with the flat Kähler metric $g_{\widetilde{X}}$ induced by $g_{T}$ and $g_{X}$ . We set $\widetilde{Y}:=p^{-1}(Y)$ .

Let $\widetilde{E}$ be the pull back of $E$ with the holomorphic structure $p^{\ast}(\overline{\partial}_{E})+p^{\ast}(f)\,d\overline{z}$ . According to the dimensional reduction of Hitchin, a Hermitian metric $h$ of $E_{|Y}$ is a harmonic metric of $(E,\overline{\partial}_{E},\theta)_{|Y}$ if and only if $\Lambda_{\widetilde{Y}}R(p^{\ast}h)=0$ . According to a theorem of Donaldson [9], there exists a unique Hermitian metric $\widetilde{h}$ of $\widetilde{E}$ such that $\Lambda_{\widetilde{Y}}R(\widetilde{h})=0$ and that $\widetilde{h}_{|\partial Y}=p^{\ast}(h_{\partial Y})$ . By the uniqueness, $\widetilde{h}$ is $T$ -invariant. Hence, there uniquely exists a harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)_{|Y}$ which induces $\widetilde{h}$ . It satisfies $h_{|\partial Y}=h_{\partial Y}$ . ∎

Let $h_{0}$ be a Hermitian metric of $E$ . Assume that $\det(h_{0})$ is flat.

Corollary 2.2.

There exists a unique harmonic metric $h$ of $E_{|Y}$ such that $h_{|\partial Y}=h_{0|\partial Y}$ and that $\det(h)=\det(h_{0})_{|Y}$ .

Proof There exists a unique harmonic metric $h$ such that $h_{|\partial Y}=h_{0|\partial Y}$ . We obtain $\det(h)_{|\partial Y}=\det(h_{0})_{|\partial Y}$ . Note that both $\det(h)$ and $\det(h_{0})_{|Y}$ are flat. By the uniqueness in Proposition 2.1, we obtain $\det(h)=\det(h_{0})_{|Y}$ . ∎

2.1.1. Homogeneous case

Let $G$ be a compact Lie group. Let $\kappa:G\longrightarrow S^{1}$ be a character. Suppose that $X$ is equipped with a $G$ -action, which can be trivial. A Higgs bundle $(E,\overline{\partial}_{E},\theta)$ is called $(G,\kappa)$ -homogeneous if $(E,\overline{\partial}_{E})$ is $G$ -equivariant, and $g^{\ast}(\theta)=\kappa(g)\theta$ for any $g\in G$ .

Lemma 2.3.

If $Y$ and $h_{\partial Y}$ are $G$ -invariant in Proposition 2.1, the harmonic metric $h$ is also $G$ -invariant.

Proof For any $g\in G$ , $g^{\ast}h$ is a harmonic metric of $(E,\overline{\partial}_{E},\kappa(g)\theta)$ such that $(g^{\ast}h)_{|\partial Y}=h_{\partial Y}$ . Because $|\kappa(g)|=1$ , $g^{\ast}(h)$ is also a harmonic metric of $(E,\overline{\partial}_{E},\theta)$ . By the uniqueness, we obtain $g^{\ast}(h)=h$ . ∎

2.1.2. Donaldson functional

Let $\mathcal{H}(E_{|Y},h_{\partial Y})$ be the space of $C^{\infty}$ -Hermitian metrics $h$ of $E_{|Y}$ such that $h_{|\partial Y}=h_{\partial Y}$ . For two metrics $h_{1},h_{2}\in\mathcal{H}(E_{|Y},h_{\partial Y})$ , let $u$ be the automorphism of $E$ which is self-adjoint with respect to both $h_{i}$ , determined by $h_{2}=h_{1}e^{u}$ . Then, we define

M(h_{1},h_{2}):=\sqrt{-1}\int_{Y}\mathop{\rm Tr}\nolimits(u\Lambda F(h_{1}))+\int_{Y}h_{1}\Bigl{(}\Psi(u)\bigl{(}(\overline{\partial}_{E}+\theta)u\bigr{)},(\overline{\partial}_{E}+\theta)u\Bigr{)}.

(See §2.4 below for $F(h_{1})$ . See [38, §4] for $\Psi(u)$ .)

Lemma 2.4.

Let $h_{0}$ be any element of $\mathcal{H}(E_{|Y},h_{\partial Y})$ . Let $h$ be the harmonic metric as in Proposition 2.1. Then, we obtain $M(h_{0},h)\leq 0$ .

Proof By the dimensional reduction as in the proof of Proposition 2.1, the claim is reduced to [33, Proposition 2.18]. ∎

2.2. Convergence

Let $X$ be an open Riemann surface. Let $G$ be a compact Lie group acting on $X$ . Let $\kappa:G\longrightarrow S^{1}$ be a character. Let $(E,\overline{\partial}_{E},\theta)$ be a $(G,\kappa)$ -homogeneous Higgs bundle on $X$ . Let $h_{0}$ be any $G$ -invariant Hermitian metric of $E$ .

Definition 2.5.

An exhaustive family $\{X_{i}\}$ of a Riemann surface $X$ means an increasing sequence of relatively compact $G$ -invariant open subsets $X_{1}\subset X_{2}\subset\cdots$ of $X$ such that $X=\bigcup X_{i}$ . The family is called smooth if $\partial X_{i}$ are smooth.

Let $\{X_{i}\}$ be a smooth exhaustive family of $X$ . The restriction $h_{0|X_{i}}$ is denoted by $h_{0,i}$ . Let $h_{i}$ $(i=1,2,\ldots)$ be $G$ -invariant harmonic metrics of $(E,\overline{\partial}_{E},\theta)_{|X_{i}}$ . Let $s_{i}$ be the automorphism of $E_{|X_{i}}$ determined by $h_{i}=h_{0,i}\cdot s_{i}$ . Let $f$ be an ${\mathbb{R}}_{>0}$ -valued function on $X$ such that each $f_{|X_{i}}$ is bounded.

Proposition 2.6.

Assume that $|s_{i}|_{h_{0,i}}+|s^{-1}_{i}|_{h_{0,i}}\leq f_{|X_{i}}$ for any $i$ . Then, there exists a subsequence $s_{i(j)}$ which is convergent to a $G$ -invariant automorphism $s_{\infty}$ of $E$ on any relatively compact subset of $X$ in the $C^{\infty}$ -sense. As a result, we obtain a $G$ -invariant harmonic metric $h_{\infty}=h_{0}s_{\infty}$ of $(E,\overline{\partial}_{E},\theta)$ as the limit of the subsequence $h_{i(j)}$ . Moreover, we obtain $|s_{\infty}|_{h_{0}}+|s_{\infty}^{-1}|_{h_{0}}\leq f$ . In particular, if $f$ is bounded, $h_{0}$ and $h_{\infty}$ are mutually bounded.

Proof We explain an outline of the proof. Let $g_{X}$ be a Kähler metric of $X$ . According to a general formula (19) below, the following holds on any $X_{i}$ :

\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s_{i})=-\mathop{\rm Tr}\nolimits\bigl{(}s_{i}\Lambda F(h_{0,i})\bigr{)}-\bigl{|}(\overline{\partial}+\theta)(s_{i})\cdot s_{i}^{-1/2}\bigr{|}^{2}_{h_{0,i},g}.

(12)

Let $K$ be any compact subset of $X$ . Let $N$ be a relatively compact neighbourhood of $K$ in $X$ . Let $\chi:X\longrightarrow{\mathbb{R}}_{\geq 0}$ be a $C^{\infty}$ -function such that (i) $\chi_{|K}=1$ , (ii) $\chi_{|X\setminus N}=0$ , (iii) $\chi^{-1/2}\partial\chi$ and $\chi^{-1/2}\overline{\partial}\chi$ on $\{P\in X\,|\,\chi(P)>0\}$ induces a $C^{\infty}$ -function on $X$ .

There exist $i_{0}$ such that $N$ is a relatively compact open subset of $X_{i}$ for any $i\geq i_{0}$ . We obtain the following:

\sqrt{-1}\Lambda\overline{\partial}\partial(\chi\mathop{\rm Tr}\nolimits(s_{i}))=\chi\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s_{i})+(\sqrt{-1}\Lambda\overline{\partial}\partial\chi)\cdot\mathop{\rm Tr}\nolimits(s_{i})\\ +\sqrt{-1}\Lambda(\overline{\partial}\chi\partial\mathop{\rm Tr}\nolimits(s_{i}))-\sqrt{-1}\Lambda(\partial\chi\overline{\partial}\mathop{\rm Tr}\nolimits(s_{i})).

(13)

Note that $|\overline{\partial}_{E}s_{i}|_{h_{i},g_{X}}=|\partial_{E,h_{i}}s_{i}|_{h_{i},g_{X}}$ , and that

\left|\int_{X}\sqrt{-1}\Lambda(\overline{\partial}\chi\partial\mathop{\rm Tr}\nolimits(s_{i}))\right|\leq\left(\int_{X}|\chi^{-1/2}\overline{\partial}\chi|^{2}\right)^{1/2}\left(\int_{X}\chi|\partial_{E,h_{i}}s_{i}|^{2}_{h_{0},g_{X}}\right)^{1/2}\!\!\!\!.

(14)

Note that there exists $C_{0}>0$ such that $|s_{i}|_{h_{0}}+|s_{i}^{-1}|_{h_{0}}\leq C_{0}$ on $N$ for any $i$ . By (12), (13) and (14), there exist $C_{j}>0$ $(j=1,2)$ such that the following holds for any sufficiently large $i$ :

\int\chi\bigl{|}\overline{\partial}_{E}s_{i}\bigr{|}^{2}_{h_{0},g_{X}}+\int\chi\bigl{|}[\theta,s_{i}]\bigr{|}^{2}_{h_{0},g_{X}}\leq\\ C_{1}+C_{2}\left(\int\chi\bigl{|}\overline{\partial}_{E}s_{i}\bigr{|}^{2}_{h_{0},g_{X}}+\int\chi\bigl{|}[\theta,s_{i}]\bigr{|}^{2}_{h_{0},g_{X}}\right)^{1/2}.

(15)

Therefore, there exists $C_{3}>0$ such that the following holds for any sufficiently large $i$ :

\int\chi\bigl{|}\overline{\partial}_{E}s_{i}\bigr{|}^{2}_{h_{0},g_{X}}+\int\chi\bigl{|}[\theta,s_{i}]\bigr{|}^{2}_{h_{0},g_{X}}\leq C_{3}.

We obtain the boundedness of the $L^{2}$ -norms of $\overline{\partial}_{E}s_{i}$ and $\partial_{E,h_{i}}s_{i}$ $(i\geq i_{0})$ on $K$ with respect to $h_{0}$ and $g_{X}$ . By a variant of Simpson’s main estimate (see [32, Proposition 2.1]), we obtain the boundedness of the sup norms of $\theta$ on $N$ with respect to $h_{i}$ and $g_{X}$ . By the Hitchin equation, we obtain the boundedness of the sup norms of $\overline{\partial}_{E}(s_{i}^{-1}\partial_{E,h_{i}}s_{i})$ on $N$ with respect to $h_{i}$ and $g_{X}$ . By using the elliptic regularity, we obtain that the $L_{1}^{p}$ -norms of $s_{i}^{-1}\partial_{E,h_{i}}(s_{i})$ on a relatively compact neighbourhood of $K$ are bounded for any $p>1$ . It follows that $L_{2}^{p}$ -norms of $s_{i}$ on a relatively compact neighbourhood of $K$ are bounded for any $p$ . Hence, a subsequence of $s_{i}$ is weakly convergent in $L_{2}^{p}$ on a relatively compact neighbourhood of $K$ . By the bootstrapping argument using a general formula (18) below, we obtain that the sequence is convergent on a relatively compact neighbourhood of $K$ in the $C^{\infty}$ -sense. By using the diagonal argument, we obtain that a subsequence of $s_{i}$ is weakly convergent in $C^{\infty}$ -sense on any compact subset. ∎

Let us give a complement to Proposition 2.6.

Proposition 2.7.

Suppose that $h_{i|\partial X_{i}}=h_{0|\partial X_{i}}$ and that $\det(h_{0})$ is flat. Then, in Proposition 2.6, we obtain $\det(s_{\infty})=1$ . Moreover, if $\int_{X}|F(h_{0})|_{h_{0},g_{X}}<\infty$ , and if $f$ is bounded, then $|(\overline{\partial}_{E}+\theta)(s_{\infty})|_{h_{0},g_{X}}$ is $L^{2}$ , where $g_{X}$ denotes any Kähler metric of $X$ .

Proof Because $\det(s_{i})=1$ , the first claim is clear. Let $g_{X}$ be any Kähler metric of $X$ . Note that $\mathop{\rm Tr}\nolimits(s_{i})\geq\mathop{\rm rank}\nolimits(E)$ . We obtain the following by using Green formula:

\int_{X_{i}}\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s_{i})\geq 0.

(16)

We obtain the following from (12) and (16):

\int_{X_{i}}\bigl{|}(\overline{\partial}_{E}+\theta)(s_{i})\cdot s_{i}^{-1/2}\bigr{|}^{2}_{h_{0},g_{X}}\leq-\int_{X_{i}}\sqrt{-1}\mathop{\rm Tr}\nolimits\bigl{(}s_{i}\Lambda F(h_{0})\bigr{)}.

We obtain the claim of the corollary by Fatou’s lemma. ∎

Remark 2.8.

In [34] and [33], a Hermitian-Einstein metric of a holomorphic vector bundle $(E,\overline{\partial}_{E})$ on a Kähler manifold $Y$ is constructed as a limit of Hermitian-Einstein metrics $(E,\overline{\partial}_{E})_{|Y_{i}}$ for an exhaustive family $Y_{i}$ of $Y$ . For the proof of Proposition 2.6, we may also apply an argument in [34] and [33, §2.8] by using the dimensional reduction.

2.3. A Kobayashi-Hitchin correspondence

Let $X$ be an open Riemann surface. Let $g_{X}$ be any Kähler metric of $X$ . Let $(E,\overline{\partial}_{E},\theta)$ be a $(G,\kappa)$ -homogeneous Higgs bundle on $X$ . Let $h_{0}$ be a $G$ -invariant Hermitian metric of $E$ such that $\det(h_{0})$ is flat.

Condition 2.9.

The support of $F(h_{0})$ is compact.

The condition clearly implies that $|F(h_{0})|$ is integrable on $X$ . For any Higgs subbundle $E^{\prime}$ of $E$ , let $h_{0}^{\prime}$ be the induced metric of $E^{\prime}$ , and we set

\deg^{\mathop{\rm an}\nolimits}(E^{\prime},h_{0}):=\int\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits F(h_{0}^{\prime}).

Recall the Chern-Weil formula [38, Lemma 3.2]:

\deg^{\mathop{\rm an}\nolimits}(E^{\prime},h_{0})=\int\sqrt{-1}\mathop{\rm Tr}\nolimits(p_{E^{\prime}}\Lambda F(h_{0}))-\int\bigl{|}(\overline{\partial}_{E}+\theta)p_{E^{\prime}}\bigr{|}^{2}_{h_{0},g_{X}},

where $p_{E^{\prime}}$ denote the orthogonal projection of $E$ onto $E^{\prime}$ . By Condition 2.9, $\deg^{\mathop{\rm an}\nolimits}(E^{\prime},h_{0})$ is well defined in ${\mathbb{R}}\cup\{-\infty\}$ . Because $\det(h_{0})$ is assumed to be flat, we obtain $\mathop{\rm Tr}\nolimits(F(h_{0}))=0$ , and hence $\deg^{\mathop{\rm an}\nolimits}(E,h_{0})=0$ .

Definition 2.10.

$(E,\overline{\partial}_{E},\theta,h_{0})$ is analytically stable with respect to the $G$ -action if $\deg^{\mathop{\rm an}\nolimits}(E^{\prime},h_{0})<0$ for any proper $G$ -equivariant Higgs subbundle $E^{\prime}$ .

Proposition 2.11.

If $(E,\overline{\partial}_{E},\theta,h_{0})$ is analytically stable with respect to the $G$ -action, there exists a $G$ -invariant harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)$ such that (i) $h_{0}$ and $h$ are mutually bounded, (ii) $\det(h)=\det(h_{0})$ , (iii) $(\overline{\partial}_{E}+\theta)(h\cdot h_{0}^{-1})$ is $L^{2}$ .

Proof Let $N_{0}$ be a relatively compact neighbourhood of the support of $F(h_{0})$ . Let $N_{1}$ be a relatively compact neighbourhood of $\overline{N}_{0}$ . Let $X_{1}\subset X_{2}\subset\cdots$ be a smooth exhaustive sequence of $X$ such that $\overline{N}_{1}\subset X_{1}$ . We set $h_{0,i}:=h_{0|X_{i}}$ .

There exists a harmonic metric $h_{i}$ of $(E,\overline{\partial}_{E},\theta)_{|X_{i}}$ such that $h_{i|\partial X_{i}}=h_{0|\partial X_{i}}$ . Let $s_{i}$ be the automorphism of $E_{|X_{i}}$ determined by $h_{i}=h_{0,i}\cdot s_{i}$ . Note that $\det(s_{i})=1$ . According to the inequality (21) below, we obtain

\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm Tr}\nolimits(s_{i})\leq\bigl{|}F(h_{0,i})\bigr{|}_{h_{0},g_{X}}.

(17)

Note that $\mathop{\rm Tr}\nolimits s_{i|\partial X_{i}}=\mathop{\rm rank}\nolimits(E)$ and that $F(h_{0,i})=0$ on $X_{i}\setminus N_{0}$ . By (17), we obtain the following

\max_{X_{i}}\log\mathop{\rm Tr}\nolimits(s_{i})\leq\max\Bigl{\{}\max_{N_{0}}\log\mathop{\rm Tr}\nolimits(s_{i}),\mathop{\rm rank}\nolimits(E)\Bigr{\}}.

Let $\|\log\mathop{\rm Tr}\nolimits(s_{i})\|_{h_{0},N_{1}}$ denote the $L^{1}$ -norm of $\log\mathop{\rm Tr}\nolimits(s_{i})_{|N_{1}}$ with respect to $h_{0}$ and $g_{X}$ . By using the argument in the proof of [38, Proposition 2.1] together with (17), we obtain $C_{j}>0$ $(j=0,1)$ depending only on $|F(h_{0})|_{h_{0},g_{X}}$ such that

\max_{N_{0}}|\log\mathop{\rm Tr}\nolimits(s_{i})|_{h_{0}}\leq C_{0}\|\log\mathop{\rm Tr}\nolimits(s_{i})\|_{h_{0},N_{1}}+C_{1}.

Let $u_{i}$ be the endomorphism of $E_{|X_{i}}$ which is self-adjoint with respect to $h_{0,i}$ and $h_{i}$ determined by $s_{i}=e^{u_{i}}$ . There exists $C_{j}$ $(j=2,3)$ depending only on $\mathop{\rm rank}\nolimits(E)$ such that

|u_{i}|_{h_{0}}\leq C_{2}(\log|s_{i}|_{h_{0}}+1),\quad\quad\log|s_{i}|_{h_{0}}\leq C_{3}(|u_{i}|_{h_{0}}+1).

Let $\|u_{i}\|_{h_{0},N_{1}}$ denote the $L^{1}$ -norm of $u_{i|N_{1}}$ with respect to $h_{0}$ and $g_{X}$ . There exists $C_{j}$ $(j=4,5)$ , depending only on $\mathop{\rm rank}\nolimits(E)$ and $|F(h_{0})|$ such that

\max_{X_{i}}|u_{i}|_{h_{0}}\leq C_{4}\|u_{i}\|_{h_{0},N_{1}}+C_{5}.

The rest is almost the same as the proof of [33, Theorem 2.5], particularly [33, Proposition 2.32] and [33, §2.7.4]. We explain only an outline.

Suppose that $\sup_{X_{i}}|s_{i}|_{h_{0,i}}\to\infty$ as $i\to\infty$ . It implies $\|u_{i}\|_{h_{0},N_{1}}\to\infty$ . We set $v_{i}:=u_{i}/\|u_{i}\|_{h_{0},N_{1}}$ . They are endomorphisms of $E_{|X_{i}}$ which are self adjoint with respect to $h_{i}$ and $h_{0,i}$ . We can prove the following lemma by using the argument in [38, Lemma 5.4].

Lemma 2.12.

There exists a $G$ -invariant $L_{1}^{2}$ -section $v_{\infty}$ of $\mathop{\rm End}\nolimits(E)$ on $X$ such that the following holds.

•

$v_{\infty}\neq 0$ .
•

A subsequence of $\{v_{i}\}$ is weakly convergent to $v_{\infty}$ in $L_{1}^{2}$ on any compact subset of $X$ .

•

Let $\Phi:{\mathbb{R}}\times{\mathbb{R}}\longrightarrow{\mathbb{R}}_{>0}$ be a $C^{\infty}$ -function such that $\Phi(y_{1},y_{2})<(y_{1}-y_{2})^{-1}$ if $y_{1}>y_{2}$ . Then, the following holds:

\sqrt{-1}\int_{X}\mathop{\rm Tr}\nolimits\bigl{(}v_{\infty}\Lambda F(h_{0})\bigr{)}+\int_{X}h_{0}\Bigl{(}\Phi(v_{\infty})\bigl{(}(\overline{\partial}_{E}+\theta)v_{\infty}\bigr{)},(\overline{\partial}_{E}+\theta)v_{\infty}\Bigr{)}\leq 0.

(See [38, §4] for $\Phi(v_{\infty})$ .) ∎

By applying Lemma 2.12 with the argument in the proof of [38, Lemma 5.5], we obtain that the eigenvalues of $v_{\infty}$ are constant. Let $\lambda_{1}<\cdots<\lambda_{\ell}$ be the eigenvalues of $v_{\infty}$ . As the orthogonal projections onto the sum of the eigen spaces associated with $\lambda_{1},\ldots,\lambda_{i}$ , we obtain an $L_{1}^{2}$ -section $\pi_{i}$ of $\mathop{\rm End}\nolimits(E)$ for which $\pi_{i}^{2}=\pi_{i}$ and $(\mathop{\rm id}\nolimits_{E}-\pi_{i})\circ(\overline{\partial}_{E}+\theta)\pi_{i}=0$ . (See [38, §4] for more precise construction of $\pi_{i}$ .) According to the regularity of Higgs $L_{1}^{2}$ -subbundle [38, Proposition 5.8], there exists a Higgs subbundle $E_{i}\subset E$ such that the orthogonal projection onto $E_{i}$ is equal to $\pi_{i}$ . By the argument in the proof of [38, Proposition 5.3], we obtain $\deg(E_{i},h_{0})>0$ for one of $i$ , which contradicts with the stability condition. Hence, there exists $C>0$ , which is independent of $i$ , such that $|s_{i}|_{h_{0,i}}<C$ . Then, the claim of Proposition 2.11 follows from Proposition 2.6. ∎

2.4. Appendix

We recall some fundamental formulas due to Simpson [38, Lemma 3.1] for the convenience of the readers. Let $(E,\overline{\partial}_{E},\theta)$ be a Higgs bundle on a Riemann surface $X$ . For a Hermitian metric $h$ of $E$ , we obtain the Chern connection $\nabla_{h}=\overline{\partial}_{E}+\partial_{E,h}$ of $(E,\overline{\partial}_{E},h)$ . The curvature of $\nabla_{h}$ is denoted by $R(h)$ . We also obtain the adjoint $\theta^{\dagger}_{h}$ of $\theta$ with respect to $h$ . The curvature of $\nabla_{h}+\theta+\theta_{h}^{\dagger}$ is denoted by $F(h)$ , i.e., $F(h)=R(h)+[\theta,\theta^{\dagger}_{h}]$ .

Let $h_{i}$ $(i=1,2)$ be Hermitian metrics of $E$ . We obtain the automorphism $s$ of $E$ determined by $h_{2}=h_{1}\cdot s$ . Let $g$ be a Kähler metric of $X$ , let $\Lambda$ denote the adjoint of the multiplication of the associated Kähler form. Then, according to [38, Lemma 3.1 (a)], we obtain the following on $X$ :

\sqrt{-1}\Lambda\bigl{(}\overline{\partial}_{E}+\theta\bigr{)}\circ\bigl{(}\partial_{E,h_{1}}+\theta^{\dagger}_{h_{1}}\bigr{)}s=\\ s\sqrt{-1}\Lambda\bigl{(}F(h_{2})-F(h_{1})\bigr{)}+\sqrt{-1}\Lambda\Bigl{(}\bigl{(}\overline{\partial}_{E}+\theta\bigr{)}(s)s^{-1}\bigl{(}\partial_{E,h_{1}}+\theta^{\dagger}_{h_{1}}\bigr{)}(s)\Bigr{)}.

(18)

By taking the trace, and by using [38, Lemma 3.1 (b)], we obtain

\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s)=\sqrt{-1}\mathop{\rm Tr}\nolimits\Bigl{(}s\Lambda\bigl{(}F(h_{2})-F(h_{1})\bigr{)}\Bigr{)}-\Bigl{|}\bigl{(}\overline{\partial}_{E}+\theta\bigr{)}(s)s^{-1/2}\Bigr{|}^{2}_{h_{1},g}.

(19)

Note that $(\overline{\partial}_{E}+\theta)(s)=\overline{\partial}_{E}(s)+[\theta,s]$ . Moreover, $\overline{\partial}_{E}(s)$ is a $(0,1)$ -form, and $[\theta,s]$ is a $(1,0)$ -form. Hence, (19) is also rewritten as follows:

\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s)=\sqrt{-1}\mathop{\rm Tr}\nolimits\Bigl{(}s\Lambda\bigl{(}F(h_{2})-F(h_{1})\bigr{)}\Bigr{)}\\ -\bigl{|}[\theta,s]s^{-1/2}\bigr{|}^{2}_{h_{1},g}-\bigl{|}\overline{\partial}_{E}(s)s^{-1/2}\bigr{|}_{h_{1},g}.

(20)

We also recall the following inequality [38, Lemma 3.1 (d)]:

\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm Tr}\nolimits(s)\leq\bigl{|}\Lambda F(h_{1})\bigr{|}_{h_{1}}+\bigl{|}\Lambda F(h_{2})\bigr{|}_{h_{2}}.

(21)

In particular, if both $h_{i}$ are harmonic, the functions $\mathop{\rm Tr}\nolimits(s)$ and $\log\mathop{\rm Tr}\nolimits(s)$ are subharmonic:

\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s)=-\bigl{|}(\overline{\partial}_{E}+\theta)(s)s^{-1/2}\bigr{|}^{2}_{h,g}\leq 0,\quad\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm Tr}\nolimits(s)\leq 0.

(22)

3. Preliminaries for harmonic metrics of cyclic Higgs bundles

3.1. Cyclic Higgs bundles associated with $r$ -differentials

Let $X$ be any Riemann surface equipped with a line bundle $K_{X}^{1/2}$ and an isomorphism $(K_{X}^{1/2})^{2}=K_{X}$ . We set $\mathbb{K}_{X,r}:=\bigoplus_{i=1}^{r}K_{X}^{(r+1-2i)/2}$ . We set $G_{r}:=\{a\in{\mathbb{C}}\,|\,a^{r}=1\}$ . We define the $G_{r}$ -actions on $K_{X}^{(r+1-2i)/2}$ by $a\bullet v=a^{i}v$ . They induce a $G_{r}$ -action on $\mathbb{K}_{X,r}$ .

For any holomorphic $r$ -differential $q$ on $X$ , Let $\theta(q)$ be obtained as $\theta(q)=\sum_{i=1}^{r}\theta(q)_{i}$ , where $\theta(q)_{i}:K_{X}^{(r+1-2i)/2}\longrightarrow K_{X}^{(r+1-2(i+1))/2}\otimes K_{X}$ $(i=1,\ldots,r-1)$ are induced by the identity, and $\theta(q)_{r}:K_{X}^{(-r+1)/2}\longrightarrow K_{X}^{(r-1)/2}\otimes K_{X}$ is induced by the multiplication of $q$ . Let $\mathop{\rm Harm}\nolimits(q)$ denote the set of $G_{r}$ -invariant harmonic metrics $h$ of $(\mathbb{K}_{X,r},\theta(q))$ such that $\det(h)=1$ . By the $G_{r}$ -invariance, the decomposition $\mathbb{K}_{X,r}=\bigoplus_{i=1}^{r}K_{X}^{(r+1-2i)/2}$ is orthogonal with respect to any $h\in\mathop{\rm Harm}\nolimits(q)$ , and hence we obtain the decomposition $h=\bigoplus h_{|K_{X}^{(r+1-2i)/2}}$ .

Note that $K_{X}^{(r+1-2i)/2}$ and $K_{X}^{(r+1-2(r+1-i))/2}=K_{X}^{(-r-1+2i)/2}$ are mutually dual. We say that $h\in\mathop{\rm Harm}\nolimits(q)$ is real if $h_{|K_{X}^{(r+1-2i)/2}}$ and $h_{|K_{X}^{(-r-1+2i)/2}}$ are mutually dual. Let $\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)$ denote the subset of $h\in\mathop{\rm Harm}\nolimits(q)$ which are real.

Note that $\bigl{(}K_{X}^{(r+1-2i)/2}\bigr{)}^{-1}\otimes K_{X}^{(r+1-2(i+1))/2}$ is naturally isomorphic to the tangent bundle of $X$ . Hence,

h_{|K_{X}^{(r+1-2i)/2}}^{-1}\otimes h_{|K_{X}^{(r+1-2(i+1))/2}}\quad(i=1,\ldots,r-1)

(23)

induce Kähler metrics of $X$ . If the metrics (23) induce complete distances on $X$ , $h$ is called complete.

Remark 3.1.

According to [24], there uniquely exists $h^{\mathop{\rm c}\nolimits}\in\mathop{\rm Harm}\nolimits(q)$ which is real and complete. If $q$ is nowhere vanishing on $X$ , there exists a harmonic metric $h_{\mathop{\rm can}\nolimits}$ as in [24] (see also §3.3 below).

3.2. Hermitian metrics on a vector space with a cyclic automorphism

Let $r$ be a positive integer. Let $V$ be an $r$ -dimensional ${\mathbb{C}}$ -vector space with a base ${\boldsymbol{e}}=(e_{0},\ldots,e_{r-1})$ . Let $\alpha$ be a non-zero complex number. Let $f$ be the automorphism of $V$ determined by $f(e_{i})=e_{i+1}$ $(i=0,1,\ldots,r-2)$ and $f(e_{r-1})=\alpha^{r}e_{0}$ . We put $G_{r}:=\{a\in{\mathbb{C}}\,|\,a^{r}=1\}$ . We consider the action of $G_{r}$ on $V$ determined by $a\bullet e_{i}=a^{i}e_{i}$ .

Let $\mathcal{H}(f,{\boldsymbol{e}})$ be the set of $G_{r}$ -invariant Hermitian metrics $h$ of $V$ such that $\prod_{i=0}^{r-1}h(e_{i},e_{i})=1$ . The $G_{r}$ -invariance is equivalent to the orthogonality $h(e_{i},e_{j})=0$ $(i\neq j)$ .

We put $\tau:=\exp(2\pi\sqrt{-1}/r)\in G_{r}$ . We set $v_{i}:=\sum_{j=0}^{r-1}\tau^{-ij}\alpha^{-j}e_{j}$ $(i=0,1,\ldots,r-1)$ . Then, ${\boldsymbol{v}}=(v_{0},\ldots,v_{r-1})$ is a base of $V$ such that $f(v_{i})=\tau^{i}\alpha v_{i}$ . Note that $\tau\bullet v_{i}=v_{i-1}$ $(i=1,\ldots,r-1)$ and $\tau\bullet v_{0}=v_{r-1}$ . Let us remark the following well known lemma.

Lemma 3.2.

Let $V^{\prime}\subset V$ be a $G_{r}$ -invariant ${\mathbb{C}}$ -subspace such that $f(V^{\prime})\subset V^{\prime}$ . Then, either $V^{\prime}=0$ or $V^{\prime}=V$ hold.

Proof Because $f(V^{\prime})\subset V^{\prime}$ , there is a subset $I\subset\{0,\ldots,r-1\}$ such that $V^{\prime}=\bigoplus_{i\in I}{\mathbb{C}}v_{i}$ . Because $V^{\prime}$ is $G_{r}$ -invariant, and because $G_{r}$ acts on $\{v_{0},\ldots,v_{r-1}\}$ in a cyclic way, we obtain $I=\emptyset$ or $I=\{0,\ldots,r-1\}$ . ∎

Let $h\in\mathcal{H}(f,{\boldsymbol{e}})$ . By the $G_{r}$ -invariance, $b(h):=h(v_{i},v_{i})$ is independent of $i$ . We obtain

h(e_{j},e_{j})=\frac{b(h)}{r}|\alpha|^{2j}+\frac{1}{r^{2}}\sum_{i\neq\ell}|\alpha|^{2j}\tau^{j(i-\ell)}h(v_{i},v_{\ell}).

(24)

3.2.1. Canonical metric

Let $h_{\mathop{\rm can}\nolimits}\in\mathcal{H}(f,{\boldsymbol{e}})$ be determined by

h_{\mathop{\rm can}\nolimits}(e_{j},e_{j})=|\alpha|^{-(r-1)+2j}\quad(j=0,\ldots,r-1).

(25)

Lemma 3.3.

$h_{\mathop{\rm can}\nolimits}$ is the unique Hermitian metric contained in $\mathcal{H}(f,{\boldsymbol{e}})$ such that the base ${\boldsymbol{v}}$ is orthogonal with respect to $h_{\mathop{\rm can}\nolimits}$ .

Proof We can check the orthogonality of ${\boldsymbol{v}}$ by a direct computation. Suppose that ${\boldsymbol{v}}$ is orthogonal with respect to $h_{1}\in\mathcal{H}(f,{\boldsymbol{e}})$ . By (24), we obtain

h_{1}(e_{j},e_{j})=\frac{b(h_{1})}{r}|\alpha|^{2j}.

By the condition $1=\prod_{j=0}^{r-1}h_{1}(e_{j},e_{j})$ , we obtain $b(h_{1})=r|\alpha|^{-(r-1)}$ , and hence $h_{1}=h_{\mathop{\rm can}\nolimits}$ . ∎

3.2.2. $\epsilon$ -orthogonality

Suppose $r>1$ , and let $0\leq\epsilon\leq 1$ . Suppose that the following holds for a metric $h\in\mathcal{H}(f,{\boldsymbol{e}})$ :

|h(v_{i},v_{j})|\leq\epsilon b(h)=\epsilon\cdot|v_{i}|_{h}\cdot|v_{j}|_{h}\quad(i\neq j).

The second term in the right hand side of (24) is dominated by

\frac{b(h)|\alpha|^{2j}}{r}(r-1)\epsilon.

Hence, we obtain the following description

h(e_{j},e_{j})=\frac{b(h)|\alpha|^{2j}}{r}(1+c_{j}),

where $|c_{j}|\leq(r-1)\epsilon$ and $1+c_{j}>0$ . By the condition $\prod h(e_{j},e_{j})=1$ , we obtain

1=\left(\frac{b(h)}{r}\right)^{r}|\alpha|^{r(r-1)}\prod_{j=0}^{r-1}(1+c_{j}).

In particular, we obtain $1=\left(\frac{b(h_{\mathop{\rm can}\nolimits})}{r}\right)^{r}|\alpha|^{r(r-1)}$ . Hence, we obtain

b(h_{\mathop{\rm can}\nolimits})=b(h)\cdot\prod_{j=0}^{r-1}(1+c_{j})^{1/r}.

For $0<\delta<1$ , we set $C_{\delta}:=(1-\delta)^{-1}(r-1)$ . If $(r-1)\epsilon<\delta$ , we obtain $|\log(1+c_{j})|\leq C_{\delta}\epsilon$ . For such $\epsilon$ , we obtain

\left|\log\bigl{(}b(h)/b(h_{\mathop{\rm can}\nolimits})\bigr{)}\right|=\left|\frac{1}{r}\sum_{j=0}^{r-1}\log(1+c_{j})\right|\leq C_{\delta}\epsilon.

We also obtain

\left|\log\Bigl{(}h(e_{j},e_{j})/h_{\mathop{\rm can}\nolimits}(e_{j},e_{j})\Bigr{)}\right|=\Bigl{|}\log\bigl{(}b(h)/b(h_{\mathop{\rm can}\nolimits})\bigr{)}+\log(1+c_{j})\Bigr{|}\leq 2C_{\delta}\epsilon.

(26)

3.2.3. Norm of the automorphism

For any $h\in\mathcal{H}(f,{\boldsymbol{e}})$ , we obtain $|f|_{h}\geq\sqrt{r}|\alpha|$ . Let $C$ be any positive number such that $C\geq\sqrt{r}|\alpha|$ .

Lemma 3.4.

For any $h\in\mathcal{H}(f,{\boldsymbol{e}})$ such that $|f|_{h}\leq C$ , we obtain

|e_{i}|_{h}\leq|\alpha|^{-r}C^{(r+1)/2+i}\quad(i=0,\ldots,r-2),\quad|e_{r-1}|_{h}\leq C^{(r-1)/2}.

Proof Because $e_{i+1}=f(e_{i})$ , we obtain $|e_{i+1}|_{h}\leq C|e_{i}|_{h}$ . We obtain $|e_{r-1}|_{h}\leq C^{r-1-i}|e_{i}|_{h}$ . Because $\prod|e_{i}|_{h}=1$ , we obtain

|e_{r-1}|_{h}^{r}C^{-r(r-1)/2}\leq 1,\quad\mbox{\rm i.e., }|e_{r-1}|_{h}\leq C^{(r-1)/2}.

Because $f(e_{r-1})=\alpha^{r}e_{0}$ , we obtain $|e_{0}|_{h}\leq C|\alpha|^{-r}|e_{r-1}|_{h}$ , and hence $|e_{0}|_{h}\leq|\alpha|^{-r}C^{(r+1)/2}$ . Then, we obtain $|e_{i}|_{h}\leq|\alpha|^{-r}C^{(r+1)/2+i}$ . ∎

Corollary 3.5.

We obtain

|\alpha|^{r}(C+1)^{-2r}\leq|e_{i}|_{h}\leq|\alpha|^{-r}(C+1)^{2r}.

Proof We obtain the second inequality from Lemma 3.4. By using the duality, we obtain the first inequality.

∎

Corollary 3.6.

For any $h_{i}\in\mathcal{H}(f,{\boldsymbol{e}})$ $(i=1,2)$ such that $|f|_{h_{i}}\leq C$ , we obtain

|\alpha|^{2r}(C+1)^{-4r}\leq\frac{|e_{i}|_{h_{1}}}{|e_{i}|_{h_{2}}}\leq|\alpha|^{-2r}(C+1)^{4r}.

∎

3.3. Harmonic metrics of a cyclic Higgs bundle on a disc

We shall recall variants of Simpson’s main estimate for harmonic bundles, which was pioneered in [39], and further developed in [29] and [32].

We set $U(R):=\bigl{\{}z\in{\mathbb{C}}\,\big{|}\,|z|<R\bigr{\}}$ . We set $E:=\bigoplus_{i=0}^{r-1}\mathcal{O}_{U(R)}\,e_{i}$ . Let $\beta$ be a holomorphic function on $U(R)$ . Let $f$ be the endomorphism of $E$ determined by $f(e_{i})=e_{i+1}$ $(i=0,\ldots,r-2)$ and $f(e_{r-1})=\beta e_{0}$ . We set $\theta:=f\,dz$ . We set $G_{r}:=\{a\in{\mathbb{C}}\,|\,a^{r}=1\}$ as in §3.2. We define the $G_{r}$ -action on $E$ by $a\bullet e_{i}=a^{i}e_{i}$ . Note that $a^{\ast}(\theta)=a^{-1}\theta$ . Let $\mathop{\rm Harm}\nolimits(\beta)$ denote the set of $G_{r}$ -invariant harmonic metrics of the Higgs bundle $(E,\theta)$ such that $\prod_{i=0}^{r-1}h(e_{i},e_{i})=1$ .

Proposition 3.7 ([32, Proposition 2.1]).

Fix $0<R_{1}<R$ . Then, there exist $C_{1},C_{2}>0$ depending only on $r$ , $R_{1}$ , and $R$ such that

\sup_{U(R_{1})}|f|_{h}\leq C_{1}+C_{2}\max_{U(R)}|\beta|^{1/r}

for any $h\in\mathop{\rm Harm}\nolimits(\beta)$ . ∎

Corollary 3.8.

Let $h_{0}$ be a $G_{r}$ -invariant Hermitian metric of $E$ such that $\prod_{i=0}^{r-1}h_{0}(e_{i},e_{i})=1$ . For $h\in\mathop{\rm Harm}\nolimits(\beta)$ , we obtain the automorphism $s$ of $E$ which is self-adjoint with respect to both $h$ and $h_{0}$ , determined by $h=h_{0}s$ . Let $C_{1}$ and $C_{2}$ be constants as in Proposition 3.7. Let $C_{3}>0$ be a positive constant such that $|f|_{h_{0}}\leq C_{3}$ . We set

A_{1}:=\max\Bigl{\{}C_{3},C_{1}+C_{2}\max_{z\in U(R)}|\beta(z)|^{1/r}\Bigr{\}}.

Then, the following holds at $z\in U(R_{1})$ such that $\beta(z)\neq 0$ :

\bigl{|}\log|s|_{h_{0}}(z)\bigr{|}\leq\log\Bigl{(}\sqrt{r}|\beta(z)|^{-2}(A_{1}+1)^{4r}\Bigr{)}.

Proof It follows from Corollary 3.6 and Proposition 3.7. ∎

Suppose moreover that $\beta$ is nowhere vanishing on $U(R)$ . We fix an $r$ -th root $\alpha$ of $\beta$ , i.e., $\alpha^{r}=\beta$ . We put $\tau:=\exp(2\pi\sqrt{-1}/r)$ . We set $v_{i}:=\sum_{j=0}^{r-1}\tau^{-ij}\alpha^{-j}e_{j}$ $(i=0,1,\ldots,r-1)$ . We obtain the decomposition $E=\bigoplus_{i=0}^{r-1}\mathcal{O}_{U(R)}v_{i}$ and $f(v_{i})=\tau^{i}\alpha v_{i}$ .

Proposition 3.9 ([32, Corollary 2.6]).

Assume that $|\alpha(0)-\alpha(z)|\leq\frac{1}{100}|\alpha(0)|$ on $U(R)$ . Fix $0<R_{2}<R_{1}$ . Then, there exist $C_{10}>0$ and $\epsilon_{10}>0$ depending only on $r$ , $R$ , $R_{1}$ and $R_{2}$ such that the following holds on $U(R_{2})$ for any $h\in\mathop{\rm Harm}\nolimits(\beta)$ and for $i\neq j$ :

|h(v_{i},v_{j})|\leq C_{10}\cdot\exp\Bigl{(}-\epsilon_{10}|\alpha(0)|\Bigr{)}\cdot|v_{i}|_{h}\cdot|v_{j}|_{h}.

Note that $|v_{i}|_{h}=|v_{j}|_{h}$ by the $G_{r}$ -invariance of $h$ . ∎

Let $h_{\mathop{\rm can}\nolimits}$ be the $G_{r}$ -invariant Hermitian metric determined by

h_{\mathop{\rm can}\nolimits}(e_{j},e_{j})=|\alpha|^{-(r-1)+2j}.

Then, the frame $(v_{0},\ldots,v_{r-1})$ is orthogonal, and we have

h_{\mathop{\rm can}\nolimits}(v_{i},v_{i})=r|\alpha|^{-(r-1)}.

Hence, the curvature of the Chern connection of $h_{\mathop{\rm can}\nolimits}$ is $0$ . Because $[\theta,\theta^{\dagger}_{h_{\mathop{\rm can}\nolimits}}]=0$ , $h_{\mathop{\rm can}\nolimits}$ is a harmonic metric of the Higgs bundle $(E,\theta)$ .

Corollary 3.10.

Suppose that the assumption of Proposition 3.9 is satisfied. Moreover, we assume that

(r-1)C_{10}\cdot\exp\Bigl{(}-\epsilon_{10}|\alpha(0)|\Bigr{)}\leq 1/2.

There exist $\epsilon_{i}>0$ and $C_{i}>0$ $(i=11,12,13)$ depending only on $r$ , $R$ , $R_{1}$ and $R_{2}$ such that the following holds for any $h\in\mathop{\rm Harm}\nolimits(\beta)$ on $U(R_{2})$ :

\Bigl{|}\log\Bigl{(}h(e_{j},e_{j})/h_{\mathop{\rm can}\nolimits}(e_{j},e_{j})\Bigr{)}\Bigr{|}\leq C_{11}\exp\bigl{(}-\epsilon_{11}|\alpha(0)|\bigr{)}.

(27)

\Bigl{|}\partial_{z}\log\Bigl{(}h(e_{j},e_{j})/h_{\mathop{\rm can}\nolimits}(e_{j},e_{j})\Bigr{)}\Bigr{|}\leq C_{12}\exp\bigl{(}-\epsilon_{12}|\alpha(0)|\bigr{)}.

(28)

\Bigl{|}\partial_{\overline{z}}\partial_{z}\log\Bigl{(}h(e_{j},e_{j})/h_{\mathop{\rm can}\nolimits}(e_{j},e_{j})\Bigr{)}\Bigr{|}\leq C_{13}\exp\bigl{(}-\epsilon_{13}|\alpha(0)|\bigr{)}.

(29)

Proof We obtain (27) from (26) and Proposition 3.9. Note that the Chern connection of the curvature of $h_{\mathop{\rm can}\nolimits}$ is $0$ , and that the decomposition $E=\bigoplus\mathcal{O}e_{i}$ is orthogonal with respect to $h$ and $h_{\mathop{\rm can}\nolimits}$ . Then, we obtain (29) from Proposition 3.7 and Proposition 3.9. We obtain (28) from (27) and (29) by using the elliptic regularity. ∎

3.3.1. Estimates near boundary

Let $Y$ be an open subset in $U(R)$ . Let $\overline{Y}$ denote the closure of $Y$ in $\overline{U}(R)$ . Let $h\in\mathop{\rm Harm}\nolimits(\beta_{|Y})$ such that $h$ extends to a continuous metric on $\overline{Y}$ .

Proposition 3.11.

Fix $0<R_{10}<R$ . There exists $C_{i}$ $(i=1,2,3)$ , depending only on $R_{10}$ , $R$ and $\mathop{\rm rank}\nolimits(E)$ , such that the following holds on $U(R_{10})\cap Y$ :

|f|_{h}\leq C_{1}+C_{2}\max_{z\in U(R)}|\beta(z)|^{1/r}+C_{3}\max_{U(R)\cap\partial Y}|f|_{h}.

Proof It follows from Proposition 3.13 below. ∎

Proposition 3.12.

Let $h_{0}$ be a $G_{r}$ -invariant Hermitian metric of $E$ such that $\prod_{i=0}^{r-1}h_{0}(e_{i},e_{i})=1$ . For $h\in\mathop{\rm Harm}\nolimits(\beta)$ , let $s(h)$ be the automorphism of $E_{|Y}$ which is self-adjoint with respect to $h$ and $h_{0}$ , determined by $h=h_{0}\cdot s(h)$ . Let $C_{i}$ $(i=1,2,3)$ be as in Proposition 3.11, and let $C_{4}$ be a positive constant such that $|f|_{h_{0}}<C_{4}$ . We set

A_{1}:=\max\Bigl{\{}C_{4},\,C_{1}+C_{2}\max_{z\in U(R)}|\beta(z)|^{1/r}+C_{3}\max_{U(R)\cap\partial Y}|f|_{h}\Bigr{\}}.

Then, the following holds at any $z\in U(R_{10})\cap Y$ such that $\beta(z)\neq 0$ :

\bigl{|}\log|s(h)|_{h_{0}}(z)\bigr{|}\leq\log\Bigl{(}\sqrt{r}|\beta(z)|^{-2}(A_{1}+1)^{4r}\Bigr{)}.

Proof It follows from Proposition 3.11 and Corollary 3.6. ∎

3.3.2. Appendix: A variant of Simpson’s main estimate near boundary

Let $(E,\overline{\partial}_{E},\theta)$ be a Higgs bundle defined on a neighbourhood of $\overline{U}(R)$ in ${\mathbb{C}}$ . Let $Y$ be an open subset in $U(R)$ . Let $\overline{Y}$ denote the closure of $Y$ in $\overline{U}(R)$ . Let $h$ be a harmonic metric of $(E,\overline{\partial}_{E},\theta)_{|Y}$ which extends to a continuous metric of $E_{|\overline{Y}}$ .

Let $f$ be the endomorphism of $E$ determined by $\theta=f\,dz$ . Let $M$ be a positive constant such that the following holds.

•

Let $\alpha$ be any eigenvalue of $f_{|P}$ for any $P\in\overline{U}(R)$ . Then, $|\alpha|\leq M$ .

Proposition 3.13.

Let $R_{1}<R$ . There exist positive constant $C_{i}$ $(i=1,2,3)$ , depending only on $R$ , $R_{1}$ and $\mathop{\rm rank}\nolimits(E)$ , such that the following holds on $Y\cap U(R_{1})$

|f|_{h}\leq C_{1}+C_{2}M+C_{3}\max_{U(R)\cap\partial Y}|f|_{h}.

Proof Recall the following inequality on $Y$ :

-\partial_{z}\partial_{\overline{z}}\log|f|^{2}_{h}\leq-\frac{\bigl{|}[f,f^{\dagger}_{h}]\bigr{|}^{2}}{|f|_{h}^{2}}.

For any $P\in\overline{U}(R)$ , let $\alpha_{1},\ldots,\alpha_{\mathop{\rm rank}\nolimits E}$ be the eigenvalues of $f_{|P}$ , and we set $g(P):=\sum|\alpha_{i}|^{2}$ . There exists $C_{10}>0$ such that the following holds for any $P\in Y$ :

|[f_{|P},f_{|P}^{\dagger}]|_{h}\geq C_{10}(|f_{|P}|_{h}^{2}-g(P)).

We obtain the following on $Y$ :

-\partial_{z}\partial_{\overline{z}}\log|f|^{2}_{h}(P)\leq-C_{10}^{2}\frac{(|f_{|P}|_{h}^{2}-g(P))^{2}}{|f_{|P}|_{h}^{2}}.

If $|f_{|P}|^{2}\geq 2g(P)$ , then we obtain

-\partial_{z}\partial_{\overline{z}}\log|f|^{2}_{h}(P)\leq-\frac{C_{10}^{2}}{4}|f_{|P}|^{2}_{h}.

(30)

We consider the following function on $U(R)$ :

A(z):=\frac{2R^{2}}{(R^{2}-|z|^{2})^{2}}\geq 2R^{-2}.

Note that $-\partial_{z}\partial_{\overline{z}}\log A=-A$ . Let $C_{11}$ be a constant such that $C_{11}>(C_{10}^{2}/4)^{-1}$ . We set

C_{12}:=C_{11}+R^{2}\sup_{U(R)\cap\partial Y}|f|_{h}^{2}+10\mathop{\rm rank}\nolimits(E)R^{2}M^{2}.

Note that the following holds on $U(R)$ :

-\partial_{z}\partial_{\overline{z}}\log(C_{12}A)=-A=-C_{12}^{-1}(C_{12}A)\geq-\frac{C_{10}^{2}}{4}(C_{12}A).

Let $Z$ be the set of $P\in Y$ such that $|f_{|P}|_{h}^{2}>C_{12}A(P)$ . We assume that $Z$ is non-empty, and we shall derive a contradiction. Note that $|f|_{h}^{2}<C_{12}A$ on $\partial Y\cap U(R)$ . We also note that $|f|_{h}$ is bounded around any point of $\partial Y\cap\partial U(R)$ , and that $A(z)\to\infty$ as $|z|\to R$ . Hence, $Z$ is a relatively compact open subset of $Y$ , which implies $|f|_{h}^{2}=C_{12}A$ on $\partial Z$ . For $P\in Z$ , we obtain $|f_{|P}|_{h}^{2}\geq 2g(P)$ , and hence (30) holds at $P$ . We obtain the following at $P\in Z$ :

-\partial_{z}\partial_{\overline{z}}\Bigl{(}\log|f|_{h}^{2}-\log(C_{12}A)\Bigr{)}(P)\leq-\frac{C_{10}^{2}}{4}(|f_{|P}|^{2}-C_{12}A(P))<0.

(31)

Together with $|f|_{h}^{2}=C_{12}A$ on $\partial Z$ , we obtain $|f|_{h}^{2}\leq C_{12}A$ on $Z$ , which contradicts the construction of $Z$ . Hence, we obtain that $Z=\emptyset$ . ∎

3.4. Some existence results of harmonic metrics on cyclic Higgs bundles

Let $X$ be a Riemann surface. Let $E=\bigoplus_{i=0}^{r-1}E_{i}$ be a graded holomorphic vector bundle on $X$ . Let $\theta$ be a Higgs field of $E$ such that $\theta(E_{i})\subset E_{i+1}\otimes K_{X}$ $(i=0,\ldots,r-2)$ and $\theta(E_{r-1})\subset E_{0}\otimes K_{X}$ . We assume that $\det(\theta)$ is not constantly $0$ . Assume that there exists a flat metric $h_{\det(E)}$ of $\det(E)$ . We consider the $G_{r}$ -action on $E_{i}$ given by $a\bullet v_{i}=a^{i}v_{i}$ , which induces a $G_{r}$ -action on $E$ . For any open subset $Y$ , let $\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta)_{|Y},h_{\det(E)})$ denote the set of $G_{r}$ -invariant harmonic metrics $h$ of $(E,\theta)_{|Y}$ such that $\det(h)=h_{\det(E)|Y}$ .

Remark 3.14.

For any $r$ -differential $q$ on $X$ , we obtain $\mathop{\rm Harm}\nolimits(q)=\mathop{\rm Harm}\nolimits^{G_{r}}\bigl{(}(\mathbb{K}_{X,r},\theta(q)),1\bigr{)}$ by definition.

3.4.1. Convergence

Let $X_{1}\subset X_{2}\subset\cdots$ be a smooth exhaustive family of $X$ . Suppose that $h_{i}\in\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta)_{|X_{i}},h_{\det(E)})$ . Let $h_{0}$ be any $G_{r}$ -invariant Hermitian metric of $E$ such that $\det(h_{0})=h_{\det(E)}$ . Let $s_{i}$ be the automorphism of $E_{|X_{i}}$ determined by $h_{i}=h_{0|X_{i}}s_{i}$ .

Proposition 3.15.

There exists a locally bounded function $f$ on $X$ such that $|s_{i}|_{h_{0,i}}+|s_{i}^{-1}|_{h_{0,i}}\leq f_{|X_{i}}$ for any $i$ . As a result, there exists a convergent subsequence of the sequence $\{s_{i}\}$ with the limit $s_{\infty}$ , and we obtain $h_{\infty}=h_{0}s_{\infty}\in\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta),h_{\det(E)})$ .

Proof Let $Z$ denote the zero set of $\det(\theta)$ . It is discrete in $X$ . By Corollary 3.8, for any $P\in X\setminus Z$ , there exists a relatively compact neighbourhood $X_{P}$ of $P$ in $X\setminus Z$ with a constant $C_{P}>1$ such that $C_{P}^{-1}\leq|s_{i}|_{h_{0}}\leq C_{P}$ for any large $i$ . By using the subharmonicity of $\mathop{\rm Tr}\nolimits(s_{i})$ (see §2.4), we also obtain a similar estimate locally around any point of $Z$ . Then, the claim follows from Proposition 2.6. ∎

Corollary 3.16.

Suppose that there exists a subset $Y\subset X$ and an ${\mathbb{R}}_{>0}$ -valued locally bounded function $f_{Y}$ on $Y$ such that $|s_{i|Y\cap X_{i}}|_{h_{0}}\leq f_{Y|Y\cap X_{i}}$ . Then, we obtain $|s_{\infty|Y}|_{h_{0}}\leq f_{Y}$ for $s_{\infty}$ in Proposition 3.15. ∎

3.4.2. Control of growth order

Let $g_{X}$ be any Kähler metric of $X$ . Let $h_{0}$ be a $G_{r}$ -invariant Hermitian metric of $E$ such that $\det(h_{0})=h_{\det(E)}$ . Let $\{X_{i}\}$ be a smooth exhaustive family of $X$ . Let $h_{i}\in\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta)_{|X_{i}},h_{\det(E)})$ such that $h_{i|\partial X_{i}}=h_{0|\partial X_{i}}$ .

Let $Y$ be an open subset of $X$ . Suppose the following.

•

There exists an ${\mathbb{R}}_{\geq 0}$ -valued function $A_{0}$ on a neighbourhood $N$ of $\overline{Y}$ such that the following holds on $N$ :

$\sqrt{-1}\Lambda\overline{\partial}\partial A_{0}\geq|F(h_{0})|_{h_{0},g_{X}}.$
•

There exists an ${\mathbb{R}}_{\geq 0}$ -valued harmonic function $A_{1}$ on $N$ such that $\log\mathop{\rm Tr}\nolimits(h_{i}\cdot h_{0}^{-1})\leq A_{1}+\log r$ on $X_{i}\cap\partial Y$ .

Proposition 3.17.

There exists $h\in\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta),h_{\det(E)})$ such that $\log\mathop{\rm Tr}\nolimits(h\cdot h_{0}^{-1})\leq A_{0}+A_{1}+\log r$ on $Y$ .

Proof By Proposition 3.15, we may assume that $\{h_{i}\}$ is convergent to $h_{\infty}$ . Let $s_{i}$ be the automorphism of $E_{|X_{i}}$ obtained as $s_{i}=h_{i}\cdot h_{0|X_{i}}^{-1}$ . By our choice, we obtain

\sqrt{-1}\Lambda\overline{\partial}\partial\bigl{(}\log\mathop{\rm Tr}\nolimits(s_{i})-A_{0}-A_{1}-\log r\bigr{)}\leq 0.

On $Y\cap\partial X_{i}$ , we obtain

\log\mathop{\rm Tr}\nolimits(s_{i})-A_{0}-A_{1}-\log r\leq\log\mathop{\rm Tr}\nolimits(s_{i})-\log r=0.

On $X_{i}\cap\partial Y$ , we obtain

\log\mathop{\rm Tr}\nolimits(s_{i})-A_{0}-A_{1}-\log r\leq\log\mathop{\rm Tr}\nolimits(s_{i})-A_{1}-\log r\leq 0.

Hence, we obtain $\log\mathop{\rm Tr}\nolimits(s_{i})-A_{0}-A_{1}-\log r\leq 0$ on $Y\cap X_{i}$ . We obtain $\log\mathop{\rm Tr}\nolimits(h_{\infty}h_{0}^{-1})-A_{0}-A_{1}-\log r\leq 0$ on $Y$ as in Corollary 3.16. ∎

3.4.3. Compact subsets

Proposition 3.18.

Let $K\subset X$ be a compact subset. For any $h_{1}\in\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta)_{|X\setminus K},h_{\det(E)})$ , there exists $h\in\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta),h_{\det(E)})$ such that $h_{|X\setminus N}$ and $h_{1|X\setminus N}$ are mutually bounded for any relatively compact neighbourhood $N$ of $K$ in $X$ .

Proof Let $h_{2}$ be a $G_{r}$ -invariant Hermitian metric of $E$ such that $\det(h_{2})=h_{\det(E)}$ and that $h_{2|X\setminus N}=h_{1|X\setminus N}$ for a relatively compact neighbourhood $N$ of $K$ . Note that $(E,\theta,h_{2})$ is analytically stable with respect to the $G_{r}$ -action in the sense of §2.3 because there is no proper $G_{r}$ -invariant Higgs subbundle by Lemma 3.2. (Recall that $\det(\theta)$ is not constantly $0$ .) Then, the claim follows from Proposition 2.11. ∎

3.5. Asymptotic behaviour around poles

Let $U$ be a neighbourhood of $0$ in ${\mathbb{C}}$ . We set $U^{\circ}:=U\setminus\{0\}$ . Let $q$ be a nowhere vanishing holomorphic $r$ -differential on $U^{\circ}$ which is meromorphic at $0$ . We have the expression $q=z^{m}\,\alpha\,(dz/z)^{r}$ , where $\alpha$ induces a nowhere vanishing holomorphic function on $U$ .

3.5.1. The case $m\leq 0$

For a positive number $\epsilon>0$ , we set

\rho_{\epsilon}(z):=\left\{\begin{array}[]{ll}\exp(-\epsilon|z|^{m/r})&(m<0)\\ |z|^{\epsilon}&(m=0).\end{array}\right.

There exists a relatively compact neighbourhood $U_{1}$ of $0$ in $U$ such that $q$ is nowhere vanishing on $U_{1}^{\circ}:=U_{1}\cap U^{\circ}$ . Recall that there exists the canonical harmonic metric $h_{\mathop{\rm can}\nolimits}\in\mathop{\rm Harm}\nolimits(q_{|U_{1}^{\circ}})$ .

Proposition 3.19.

Any $h\in\mathop{\rm Harm}\nolimits(q)$ is mutually bounded with $h_{\mathop{\rm can}\nolimits}$ on $U_{1}^{\circ}$ . More strongly, there exist $C>0$ and $\epsilon>0$ such that the following holds on $U_{1}^{\circ}$ :

\left|\log|(dz)^{(r+1-2i)/2}|_{h}-\log|(dz)^{(r+1-2i)/2}|_{h_{\mathop{\rm can}\nolimits}}\right|\leq C\rho_{\epsilon}.

Proof We may assume that $\{|z|\leq 1\}\subset U_{1}$ . Let $\Psi:{\mathbb{C}}\longrightarrow{\mathbb{C}}^{\ast}$ be the map given by $\Psi(\zeta)=e^{\sqrt{-1}\zeta}$ . We have the expression

\Psi^{\ast}(q)=e^{\sqrt{-1}m\zeta}\Psi^{\ast}(\alpha)(\sqrt{-1}d\zeta)^{r}.

There exists $C_{1}>1$ such that $C_{1}^{-1}\leq|\Psi^{\ast}(\alpha)|\leq C_{1}$ on $\{\mathop{\rm Im}\nolimits(\zeta)\geq 0\}$ .

Let us consider the case $m=0$ . For $\zeta_{0}$ with $y_{0}=\mathop{\rm Im}\nolimits(\zeta_{0})>0$ , let $F_{\zeta_{0}}:\{|w|<1\}\longrightarrow\{\mathop{\rm Im}\nolimits(\zeta)>0\}$ given by $F_{\zeta_{0}}(w)=y_{0}(w+\sqrt{-1})$ . We set $\widetilde{h}:=(\Psi\circ F_{\zeta_{0}})^{\ast}(h)$ and $\widetilde{h}_{\mathop{\rm can}\nolimits}:=(\Psi\circ F_{\zeta_{0}})^{\ast}(h_{\mathop{\rm can}\nolimits})$ . By Proposition 3.7 and Corollary 3.10, there exists $C_{2}>0$ and $\epsilon_{2}>0$ such that

\Bigl{|}\log|(dw)^{(r+1-2i)/2}|_{\widetilde{h}}-\log|(dw)^{(r+1-2i)/2}|_{\widetilde{h}_{\mathop{\rm can}\nolimits}}\Bigr{|}\leq C_{2}\exp(-\epsilon_{2}y_{0}).

It implies the desired estimate in the case $m=0$ .

Let us study the case $m<0$ . Take a large $T>0$ . For $\zeta_{0}$ with $y_{0}=\mathop{\rm Im}\nolimits(\zeta_{0})>0$ , we consider $F_{\zeta_{0}}:\{|w|<1\}\longrightarrow\{\mathop{\rm Im}\nolimits(\zeta)>0\}$ determined by $F_{\zeta_{0}}(w)=T(w+\zeta_{0})$ . By applying Proposition 3.7 and Corollary 3.10 to the pull back by $\Psi\circ F_{\zeta_{0}}$ , we obtain the desired estimate in the case $m<0$ . ∎

Remark 3.20.

Proposition 3.19 can be also obtained as a consequence of the classification of $G_{r}$ -equivariant good filtered Higgs bundles over a cyclic Higgs bundle on a punctured disc. See [30, §3.1.3].

3.5.2. The case $m>0$

Let $\mathcal{P}_{r,m}$ denote the set of ${\boldsymbol{b}}=(b_{i})\in{\mathbb{R}}^{r}$ satisfying

b_{1}\geq b_{2}\geq\cdots\geq b_{r}\geq b_{1}-m,\quad\sum b_{i}=-\frac{r(r+1)}{2}.

Let $\mathcal{P}^{{\mathbb{R}}}_{r,m}\subset\mathcal{P}_{r,m}$ denote the subset of ${\boldsymbol{b}}\in\mathcal{P}_{r,m}$ such that $b_{i}+b_{r+1-i}=-(r+1)$ . We obtain the following proposition from [39] (see also [30]).

Proposition 3.21.

•

For any $h\in\mathop{\rm Harm}\nolimits(q)$ , there exist ${\boldsymbol{b}}(h)=(b_{i}(h))\in\mathcal{P}_{r,m}$ determined by the following condition around $z=0$ :

\log|(dz)^{(r+1-2i)/2}|_{h}+(b_{i}(h)+i)\log|z|\\ =O\Bigl{(}\log\bigl{(}-\log|z|\bigr{)}\Bigr{)}\quad(i=1,\ldots,r).

(32)

If $h\in\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)$ , then ${\boldsymbol{b}}(h)\in\mathcal{P}^{{\mathbb{R}}}_{r,m}$ .

•

Let $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{b}}(h_{1})={\boldsymbol{b}}(h_{2})$ . Then, for any relatively compact neighbourhood $U^{\prime}$ of $0$ in $U$ , the restrictions $h_{1|U^{\prime}\setminus\{0\}}$ and $h_{2|U^{\prime}\setminus\{0\}}$ are mutually bounded.

Proof Let us explain an outline of the proof (see also [30, §3.1.2, §3.2.8]). Let $h\in\mathop{\rm Harm}\nolimits(q)$ , i.e., $h$ is a $G_{r}$ -invariant harmonic metric of $(\mathbb{K}_{U^{\circ},r},\theta(q))$ such that $\det(h)=1$ . Let $f$ be the endomorphism of $\mathbb{K}_{U^{\circ},r}$ determined by $\theta(q)=f\,dz/z$ . Because $m>0$ , the harmonic bundle $(\mathbb{K}_{U^{\circ},r},\theta(q),h)$ is tame. For any $a\in{\mathbb{R}}$ , and for any open subset $\mathcal{U}\ni 0$ , let $\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}(\mathcal{U})$ denote the space of sections $s$ of $\mathcal{U}\setminus\{0\}$ such that $|s|_{h}=O\bigl{(}|z|^{-a-\epsilon}\bigr{)}$ for any $\epsilon>0$ . For any open subset $\mathcal{U}\not\ni 0$ , let $\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}(\mathcal{U})$ denote the space of holomorphic sections of $\mathbb{K}_{U^{\circ},r}(\mathcal{U})$ . Thus, we obtain $\mathcal{O}_{U}$ -modules $\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}$ $(a\in{\mathbb{R}})$ . According to [39, Theorem 2], $\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}$ are locally free $\mathcal{O}_{U}$ -modules, and $f(\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r})\subset\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}$ for any $a$ . Because $h$ is $G_{r}$ -invariant, $\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}$ is naturally $G_{r}$ -equivariant. Hence, we obtain the decomposition

\mathcal{P}^{h}_{a}\mathbb{K}_{U^{\circ},r}=\bigoplus_{i=1}^{r}\mathcal{P}^{h}_{a}K_{U^{\circ}}^{(r+1-2i)/2}.

(33)

Here, the locally free $\mathcal{O}_{U}$ -modules $\mathcal{P}^{h}_{a}K_{U^{\circ}}^{(r+1-2i)/2}$ are obtained from $K_{U^{\circ}}^{(r+1-2i)/2}$ with $h$ as above.

Because $m>0$ , the eigenvalues of $f_{|z}$ goes to $0$ as $|z|\to 0$ . Hence, according to Simpson’s main estimate [39, Theorem 1], we obtain $|f|_{h}=O\bigl{(}(-\log|z|)^{-1}\bigr{)}$ .

We set $v_{i}:=z^{i}(dz)^{(r+1-2i)/2}$ . Then, we obtain $f(v_{i})=v_{i+1}$ $(i=1,\ldots,r-1)$ , and hence $|v_{i+1}|_{h}\leq|v_{i}|_{h}(-\log|z|)^{-1}$ . Because $\det(h)=1$ , we obtain

|v_{r}|^{r}\cdot(-\log|z|)^{r(r-1)/2}\leq\prod_{i=1}^{h}|v_{i}|_{h}=|z|^{r(r+1)/2}.

Hence, there exists $a\in{\mathbb{R}}$ such that $v_{r}$ is a section of $\mathcal{P}^{h}_{(r+1)/2}\mathbb{K}_{U^{\circ},r}$ . Because $f(v_{r})=z^{m}\alpha v_{1}$ , we obtain $|z|^{m}\cdot|\alpha|\cdot|v_{1}|_{h}\leq|v_{r}|_{h}(-\log|z|)^{-1}$ . Therefore, there exists $b\in{\mathbb{R}}$ such that $v_{i}\in\mathcal{P}^{h}_{b}\mathbb{K}_{U^{\circ},r}$ $(i=1,\ldots,r)$ . We set

b_{i}(h):=\min\bigl{\{}b\in{\mathbb{R}}\,\big{|}\,v_{i}\in\mathcal{P}^{h}_{b}\mathbb{K}_{U^{\circ},r}\bigr{\}}.

Then, by the norm estimate [39, §7] (Proposition 3.22 below), we obtain (32). Because $f(v_{i})=v_{i+1}$ and $f(v_{r})=z^{m}\alpha v_{1}$ , we obtain $b_{i}\geq b_{i+1}$ and $b_{r}\geq b_{1}-m$ . We also remark the filtered bundle $\mathcal{P}^{h}_{\ast}\mathbb{K}_{U^{\circ},r}$ is uniquely determined by the numbers ${\boldsymbol{b}}(h)=(b_{i}(h))$ because of the decomposition (33).

Let $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ such that ${\boldsymbol{b}}(h_{1})={\boldsymbol{b}}(h_{2})$ . We obtain $\mathcal{P}^{h_{1}}_{\ast}\mathbb{K}_{U^{\circ},r}=\mathcal{P}^{h_{2}}_{\ast}\mathbb{K}_{U^{\circ},r}$ . Hence, according to [39, Corollary 4.3], $h_{1}$ and $h_{2}$ are mutually bounded. ∎

Let us refine the estimate (32). For ${\boldsymbol{b}}\in\mathcal{P}_{r,m}$ , we introduce a non-negative integer $\ell$ and integers $\nu_{0}$ , $\nu_{1},\ldots,\nu_{\ell},\nu_{\ell+1}$ as follows. If $b_{1}=b_{r}$ , we set $\ell=0$ , $\nu_{0}=0$ and $\nu_{1}=r$ . If $b_{1}>b_{r}$ , we obtain the numbers $\ell\in{\mathbb{Z}}_{\geq 1}$ and $1\leq\nu_{1}<\cdots<\nu_{\ell}<r$ determined by the following condition.

•

$b_{1}=b_{\nu_{1}}$ , $b_{\nu_{j}}>b_{\nu_{j}+1}=b_{\nu_{j+1}}$ $(j=1,\ldots,\ell-1)$ , and $b_{\nu_{\ell}}>b_{\nu_{\ell}+1}=b_{r}$ .

Moreover, we set $\nu_{0}=0$ and $\nu_{\ell+1}=r$ . For $1\leq i\leq r$ , we obtain the number $j(i)$ determined by $\nu_{j(i)}<i\leq\nu_{j(i)+1}$ . If $j(i)\neq 0,\ell$ , we set

k_{i}({\boldsymbol{b}}):=\nu_{j(i)+1}-\nu_{j(i)}+1-2(i-\nu_{j(i)})=\nu_{j(i)+1}+\nu_{j(i)}+1-2i.

(34)

If $b_{1}-m\neq b_{r}$ , we define $k_{i}({\boldsymbol{b}})$ for $i$ such that $j(i)=0,\ell$ by the same formula (34). If $b_{1}-m=b_{r}$ , we define $k_{i}({\boldsymbol{b}})$ for $i$ such that $j(i)=0$ or $j(i)=\ell$ as follows:

k_{i}({\boldsymbol{b}}):=\left\{\begin{array}[]{ll}\nu_{1}+\nu_{\ell}-r+1-2i&(j(i)=0)\\ \nu_{1}+\nu_{\ell}+r+1-2i&(j(i)=\ell).\end{array}\right.

(35)

Thus, we obtain ${\boldsymbol{k}}({\boldsymbol{b}})\in{\mathbb{Z}}^{r}$ . The following is a consequence of the norm estimate of Simpson in [39].

Proposition 3.22.

For $h\in\mathop{\rm Harm}\nolimits(q)$ , we set ${\boldsymbol{k}}(h):={\boldsymbol{k}}({\boldsymbol{b}}(h))$ , where ${\boldsymbol{b}}(h)\in\mathcal{P}_{r,m}$ is given as in Proposition 3.21. Then, we obtain the following refined estimates for $i=1,\ldots,r$ :

\log|(dz)^{(r+1-2i)/2}|_{h}+(b_{i}(h)+i)\log|z|-\frac{k_{i}(h)}{2}\log\bigl{(}-\log|z|\bigr{)}=O(1).

(36)

Proof We explain an outline of the proof (see also [30]). We use the notation in the proof of Proposition 3.21. We set $\mathcal{P}^{h}_{<a}(\mathbb{K}_{U^{\circ},r})$ . For $-1<a\leq 0$ , we obtain the following finite dimensional complex vector space

\mathop{\rm Gr}\nolimits^{F}_{a}(\mathcal{P}^{h}_{0}\mathbb{K}_{U^{\circ},r|0}):=\mathcal{P}^{h}_{a}(\mathbb{K}_{U^{\circ},r})\big{/}\mathcal{P}^{h}_{<a}(\mathbb{K}_{U^{\circ},r}).

Let $W$ denote the filtration on $\mathop{\rm Gr}\nolimits^{F}_{a}(\mathcal{P}^{h}_{0}\mathbb{K}_{U^{\circ},r|0})$ obtained as the monodromy weight filtration of the nilpotent endomorphism $\mathop{\rm Res}\nolimits(\theta)$ .

Let $n_{i}(h)$ denote the integer determined by $-1<b_{i}(h)+n_{i}(h)\leq 0$ . We set $c_{i}(h):=b_{i}(h)+n_{i}(h)$ . Then, $\widetilde{v}_{i}:=z^{-n_{i}(h)}v_{i}$ is a section of $\mathcal{P}^{h}_{c_{i}(h)}(\mathbb{K}_{U^{\circ},r})$ . Let $[\widetilde{v}_{i}]$ denote the induced element of $\mathop{\rm Gr}\nolimits^{F}_{c_{i}(h)}(\mathcal{P}^{h}_{0}\mathbb{K}_{U^{\circ},r|0})$ . Then, the tuple $([\widetilde{v}_{i}]\,|\,c_{i}(h)=a)$ is a base of $\mathop{\rm Gr}\nolimits^{F}_{a}(\mathcal{P}_{0}\mathbb{K}_{U^{\circ},r|0})$ .

We obtain the following for any $i=1,\ldots,r-1$ :

\mathop{\rm Res}\nolimits(\theta)[\widetilde{v}_{i}]=\left\{\begin{array}[]{ll}[\widetilde{v}_{i+1}]&(b_{i}(h)=b_{i+1}(h))\\ 0&(b_{i}(h)>b_{i+1}(h))\end{array}\right.

(37)

We also obtain the following:

\mathop{\rm Res}\nolimits(\theta)[\widetilde{v}_{r}]=\left\{\begin{array}[]{ll}\alpha(0)[\widetilde{v}_{1}]&(b_{r}(h)=b_{1}(h)-m)\\ 0&\mbox{\rm(otherwise)}\end{array}\right.

Hence, we obtain

W_{k}\mathop{\rm Gr}\nolimits^{F}_{a}(\mathcal{P}^{h}_{0}\mathbb{K}_{U^{\circ},r|0})=\bigoplus_{\begin{subarray}{c}c_{i}(h)=a\\ k_{i}(h)\leq k\end{subarray}}{\mathbb{C}}[\widetilde{v}_{i}].

(38)

According to the norm estimate for tame harmonic bundles [39, §7], there exists $C>1$ such that

C^{-1}|z|^{-c_{i}(h)}(-\log|z|)^{k_{i}(h)/2}\leq|\widetilde{v}_{i}|_{h}\leq C|z|^{-c_{i}(h)}(-\log|z|)^{k_{i}(h)/2}.

Thus, we obtain (36). ∎

We also remark the following existence.

Proposition 3.23.

For any ${\boldsymbol{b}}\in\mathcal{P}_{r,m}$ , there exist a neighbourhood $U_{1}$ of $0$ in $U$ and $h\in\mathop{\rm Harm}\nolimits(q_{|U_{1}\cap U^{\circ}})$ such that ${\boldsymbol{b}}(h)={\boldsymbol{b}}$ .

Proof We may assume that $q$ is nowhere vanishing on $U^{\circ}$ .

Lemma 3.24.

There exist a neighbourhood $U_{1}$ of $0$ and a holomorphic function $\zeta$ on $U_{1}$ such that $\partial_{z}\zeta(0)=1$ and that $q=\alpha(0)\zeta^{m}(d\zeta/\zeta)^{r}$ . The germ of $\zeta$ at $0$ is unique.

Proof We set $\beta:=\alpha(0)^{-1}\alpha$ which is nowhere vanishing on $U$ . We have the Taylor expansion $\beta=z+\sum_{j=1}^{\infty}\beta_{j}z^{j}$ . First, let us prove that there exists a unique formal power series $\zeta^{(f)}=1+\sum_{j=2}^{\infty}\zeta^{(f)}_{j}z^{j}$ satisfying $\beta=(\zeta^{(f)}/z)^{m}(\partial_{z}\zeta^{(f)})^{r}$ . The coefficient of $z^{j}$ $(j>1)$ in $(\zeta^{(f)}/z)^{m}(\partial_{z}\zeta^{(f)})^{r}$ is described as the sum of $(m+r(j+1))\zeta^{(f)}_{j+1}$ and a polynomial of $\zeta^{(f)}_{i}$ $(i=2,\ldots,j)$ . Hence, $\zeta^{(f)}_{j+1}$ are uniquely determined by an easy induction. It particularly implies that the germ of $\zeta$ is uniquely determined. Moreover, if there exists $n\in{\mathbb{Z}}_{>0}$ such that $\beta_{j}=0$ unless $j\in n{\mathbb{Z}}_{\geq 0}$ , then we obtain $\zeta^{(f)}_{j+1}=0$ unless $j\in n{\mathbb{Z}}_{\geq 0}$ .

Let us prove the existence of a convergent solution in the case where $m_{1}:=m/r\in{\mathbb{Z}}_{>0}$ . Let $\beta^{1/r}$ be the holomorphic function on $U$ determined by the conditions $(\beta^{1/r})^{r}=\beta$ and $\beta^{1/r}(0)=1$ . For any $z\in U$ , we take a path $\gamma(z)$ connecting $0$ and $z$ in $U$ , and we set

F(z)=\int_{\gamma(z)}z^{m_{1}-1}\beta^{1/r}dz.

Then, $F(z)$ is a holomorphic function on $U$ satisfying

dF(z)=z^{m_{1}-1}\beta^{1/r}dz,\quad F(0)=0.

Note that $F(z)/z^{m_{1}}$ is holomorphic at $z=0$ , and $(F(z)/z^{m_{1}})(0)=m_{1}^{-1}$ . There exist a neighbourhood $U_{1}$ of $0$ in $U$ and a holomorphic function $\zeta$ on $U_{1}$ such that $F(z)=\frac{1}{m_{1}}\zeta^{m_{1}}$ and $\partial_{z}\zeta(0)=1$ . Then, we obtain $\zeta^{m_{1}-1}d\zeta=z^{m_{1}-1}\beta^{1/r}\,dz$ , which implies $\zeta^{m}\alpha(0)(d\zeta/\zeta)^{r}=z^{m}\alpha\cdot(dz/z)^{r}$ .

Let us study the existence in the general case. Let $\varphi_{r}:{\mathbb{C}}\longrightarrow{\mathbb{C}}$ be determined by $\varphi_{r}(w)=w^{r}$ . We set $\widetilde{U}:=\varphi_{r}^{-1}(U)$ . There exists a neighbourhood $U_{1}$ of $0$ in $U$ and a holomorphic function $\widetilde{\zeta}$ on $\varphi_{r}^{-1}(U_{1})$ such that

\varphi_{r}^{\ast}(q)=r^{r}\alpha(0)\widetilde{\zeta}^{mr}(d\widetilde{\zeta}/\widetilde{\zeta})^{r}=\alpha(0)(\widetilde{\zeta}^{r})^{m}\cdot(d\widetilde{\zeta}^{r}/\widetilde{\zeta}^{r})^{r}.

By the consideration in the first paragraph of this proof, we obtain that $\widetilde{\zeta}^{r}$ is a convergent power series of $w^{r}$ . Hence, there exists a holomorphic function $\zeta$ on $U_{1}$ such that $\varphi^{\ast}(\zeta)=\widetilde{\zeta}^{r}$ , for which $q=\zeta^{m}\alpha(0)\cdot(d\zeta/\zeta)^{r}$ holds.

∎

We obtain an embedding $\zeta:U_{1}\longrightarrow{\mathbb{C}}\subset\mathbb{P}^{1}$ . Let $w$ be the standard coordinate on ${\mathbb{C}}$ . We set $q_{0}=\alpha(0)w^{m}(dw/w)^{r}$ . We obtain $\zeta^{\ast}(q_{0})=q$ . According to [30], there exists $h_{1}\in\mathop{\rm Harm}\nolimits(q_{0})$ such that ${\boldsymbol{b}}(h_{1})={\boldsymbol{b}}$ . Then, we obtain $h=\zeta^{\ast}(h_{1})\in\mathop{\rm Harm}\nolimits(q)$ which satisfies ${\boldsymbol{b}}(h)={\boldsymbol{b}}$ . ∎

3.6. Examples on ${\mathbb{C}}$

We introduce some examples as a preparation for the proof of Theorem 5.3. We shall see more examples later in §6.3.

3.6.1. Preliminary

Let $\mathcal{P}$ denote the set of ${\boldsymbol{a}}=(a_{1},\ldots,a_{r})\in{\mathbb{R}}^{r}$ satisfying

a_{1}\geq a_{2}\geq\cdots\geq a_{r}\geq a_{1}-1,\quad\sum a_{i}=0.

For any ${\boldsymbol{a}}\in\mathcal{P}$ , we introduce a non-negative integer $\ell$ and integers $\nu_{0},\nu_{1},\ldots,\nu_{\ell},\nu_{\ell+1}$ as follows. If $a_{1}=a_{r}$ , we set $\ell=0$ , $\nu_{0}=0$ and $\nu_{1}=r$ . If $a_{1}>a_{r}$ , we obtain positive integers $\ell$ and $1\leq\nu_{1}<\cdots<\nu_{\ell}<r$ by the following condition.

•

$a_{\nu_{1}}=a_{1}$ , $a_{\nu_{j}}>a_{\nu_{j}+1}$ $(j=1,\ldots,\ell-1)$ and $a_{\nu_{\ell}}>a_{\nu_{\ell}+1}=a_{r}$ .

Moreover, we set $\nu_{0}=0$ and $\nu_{\ell+1}=r$ . For $1\leq i\leq r$ , we obtain the number $j(i)$ determined by $\nu_{j(i)}<i\leq\nu_{j(i)+1}$ . If $j(i)\neq 0,\ell$ , we set

k_{i}({\boldsymbol{a}}):=\nu_{j(i)+1}-\nu_{j(i)}+1-2(i-\nu_{j(i)})=\nu_{j(i)+1}+\nu_{j(i)}+1-2i.

(39)

If $a_{1}-1\neq a_{r}$ , we define $k_{i}({\boldsymbol{a}})$ for $i$ such that $j(i)=0,\ell$ by the same formula (39). If $a_{1}-1=a_{r}$ , we define $k_{i}({\boldsymbol{a}})$ for $i$ such that $j(i)=0$ or $j(i)=\ell$ by

k_{i}({\boldsymbol{a}}):=\left\{\begin{array}[]{ll}\nu_{1}+\nu_{\ell}-r+1-2i&(j(i)=0)\\ \nu_{1}+\nu_{\ell}+r+1-2i&(j(i)=\ell).\end{array}\right.

(40)

3.6.2. Examples

Let $z$ be the standard coordinate of ${\mathbb{C}}$ . Let $(x,y)$ be the real coordinate system obtained as $z=x+\sqrt{-1}y$ . We set $q:=\alpha e^{\sqrt{-1}z}\,(dz)^{r}$ , where $\alpha$ is a non-zero complex number.

Proposition 3.25.

For any ${\boldsymbol{a}}\in\mathcal{P}$ , there exists $h_{{\boldsymbol{a}}}\in\mathop{\rm Harm}\nolimits(q)$ such that the following estimate holds on $\{y\geq 0\}$ :

\log\bigl{|}(dz)^{(r+1-2i)/2}\bigr{|}_{h_{{\boldsymbol{a}}}}-a_{i}y-\frac{k_{i}({\boldsymbol{a}})}{2}\log(y+2)=O(1).

Proof We apply the Kobayashi-Hitchin correspondence on ${\mathbb{C}}^{\ast}$ to construct such $h\in\mathop{\rm Harm}\nolimits(q)$ in [30]. Let $\varphi:{\mathbb{C}}\longrightarrow{\mathbb{C}}^{\ast}$ be determined by $\varphi(z)=-\sqrt{-1}re^{\sqrt{-1}z/r}$ . We obtain $\varphi^{\ast}((dw)^{r})=e^{\sqrt{-1}z}(dz)^{r}$ .

We consider the $r$ -differential $q_{1}:=\alpha(dw)^{r}$ on ${\mathbb{C}}^{\ast}$ . For any ${\boldsymbol{a}}\in\mathcal{P}$ , we define ${\boldsymbol{b}}\in\mathcal{P}_{r,r}$ by the following relation:

a_{i}=\frac{1}{r}\left(b_{i}+\frac{r+1}{2}\right).

Note that ${\boldsymbol{k}}({\boldsymbol{a}})={\boldsymbol{k}}({\boldsymbol{b}})$ . According to [30], there exists $h_{1}\in\mathop{\rm Harm}\nolimits(q_{1})$ such that

\log|(dw)^{(r+1-2i)/2}|_{h_{1}}+(b_{i}+i)\log|w|-\frac{k_{i}({\boldsymbol{b}})}{2}\log\bigl{(}-\log|w|\bigr{)}=O(1)

around $w=0$ . We set $h_{{\boldsymbol{a}}}:=\varphi^{-1}(h_{1})\in\mathop{\rm Harm}\nolimits(q)$ . Note that

\varphi^{\ast}((dw)^{(r+1-2i)/2})=e^{\frac{r+1-2i}{2r}\sqrt{-1}z}(dz)^{(r+1-2i)/2}.

Hence, we obtain the following estimates for $i=0,\ldots,r-1$ as $y\to\infty$ :

\log|(dz)^{(r+1-2i)/2}|_{h_{{\boldsymbol{a}}}}-\left(b_{i}+\frac{r+1}{2}\right)\frac{y}{r}-\frac{k_{i}({\boldsymbol{a}})}{2}\log(y+2)=O(1).

Because $h_{{\boldsymbol{a}}}$ is invariant under the translation by $(x,y)\longmapsto(x+2\pi,y)$ , we obtain the desired estimate for $h_{{\boldsymbol{a}}}$ . ∎

Remark 3.26.

Later in §6.3, we shall see that the harmonic metric $h_{{\boldsymbol{a}}}$ is uniquely determined by the condition

\log\bigl{|}(dz)^{(r+1-2i)/2}\bigr{|}_{h_{{\boldsymbol{a}}}}-a_{i}y=O\bigl{(}\log(2+y)\bigr{)}

on $\{y\geq 0\}$ . Moreover, for any $h\in\mathop{\rm Harm}\nolimits(q)$ , there exists ${\boldsymbol{a}}\in\mathop{\rm Harm}\nolimits(q)$ such that $h=h_{{\boldsymbol{a}}}$ .

Remark 3.27.

The examples in Proposition 3.25 will be used in the proof of Lemma 5.25.

4. Preliminary from the classical analysis

4.1. Subharmonic functions on sectors

4.1.1. Phragmén-Lindelöf theorem for subharmonic functions

For any $1/2<a$ and $0\leq R$ , we set $D_{a}(R):=\bigl{\{}z\in{\mathbb{C}}\,\big{|}\,|z|>R,\,|\arg(z)|<\pi/2a\bigr{\}}$ . Let $\overline{D}_{a}(R)$ denote the closure in ${\mathbb{C}}$ .

Proposition 4.1 ([22, §7.3, Theorem 3]).

Let $f$ be a continuous function on $\overline{D}_{a}(0)$ which is subharmonic in $D_{a}(0)$ . Assume that there exist $C>0$ , $0<\rho<a$ , $M\in{\mathbb{R}}$ and an exhaustive family $\{K_{i}\}$ of $D_{a}(0)$ such that the following holds.

•

$f\leq C(|z|^{\rho}+1)$ on $\partial K_{i}$ .
•

$f\leq M$ on $\partial D_{a}(0)$ .

Then, $f\leq M$ on $D_{a}(0)$ .

Proof There exists $C_{1}>0$ such that $f\leq C_{1}\mathop{\rm Re}\nolimits\bigl{(}z^{\rho}+1\bigr{)}$ on $\partial K_{i}$ . Because $f$ is assumed to be subharmonic, we obtain that $f\leq C_{1}\mathop{\rm Re}\nolimits\bigl{(}z^{\rho}+1\bigr{)}$ on $K_{i}$ , and hence on $D_{a}(0)$ . There exists $C_{2}>0$ such that $f\leq C_{2}(|z|^{\rho}+1)$ on $D_{a}(0)$ . Then, the claim follows from Phragmén-Lindelöf theorem for subharmonic functions (see [22, §7.3, Theorem 3]). ∎

Corollary 4.2.

Let $f$ be a subharmonic $C^{\infty}$ -function on a neighbourhood of $\overline{D}_{a}(R)$ . Assume the following.

•

$f$ is bounded from above on $\partial\overline{D}_{a}(R)$ .
•

There exist $C>0$ , $0<\rho<a$ and an exhaustive family $\{K_{i}\}$ of $D_{a}(R)$ such that $f\leq C(|z|^{\rho}+1)$ on $\partial K_{i}$ .

Then, $f$ is bounded from above on $\overline{D}_{a}(R)$ .

Proof There exists an ${\mathbb{R}}$ -valued $C^{\infty}$ -function $g$ on a neighbourhood of $\overline{D}_{a}(0)$ such that $g=f$ on $\overline{D}_{a}(R+1)$ . Let $\chi$ be a $C^{\infty}$ -function on $\overline{D}_{a}(0)$ with compact support such that (i) $0\leq\chi\leq 1$ , (ii) $\chi(z)=1$ on $\bigl{\{}z\in\overline{D}_{a}(0)\,\big{|}\,|z|\leq R+1\bigr{\}}$ . Note that $(1-\chi)\triangle g\geq 0$ on $\overline{D}_{a}(0)$ , where $\triangle=4\partial_{z}\partial_{\overline{z}}$ . There exists a continuous bounded function $G$ on $\overline{D}_{a}(0)$ such that $\triangle G=\chi\triangle g$ and that $G_{|\partial D_{a}(0)}=0$ . There exists $C_{1}>0$ such that $g-G\leq C_{1}$ on $\partial\overline{D}_{a}(0)$ . There exists $C_{2}>0$ such that $g-G\leq C_{2}\mathop{\rm Re}\nolimits\bigl{(}z^{\rho}+1\bigr{)}$ on $\partial K_{i}$ . Because $g-G$ is subharmonic, we obtain that $g-G_{1}\leq C_{2}\mathop{\rm Re}\nolimits\bigl{(}z^{\rho}+1\bigr{)}$ on $K_{i}$ , and hence on $D_{a}(R)$ . Because $D_{a}(0)\setminus D_{a}(R)$ is relatively compact, there exists $C_{3}>0$ such that $g-G_{1}\leq C_{3}(|z|^{\rho}+1)$ on $D_{a}(0)$ . By the Phragmén-Lindelöf theorem, we obtain the desired boundedness. ∎

Let us give a variant. Let $b_{+}$ , $b_{-}$ and $\kappa$ be non-negative numbers such that $b:=\max\{b_{+},b_{-}\}\leq\kappa$ .

Corollary 4.3.

Let $f$ be a subharmonic function on $D_{a}(R)$ which extends to a continuous function on $\overline{D}_{a}(R)$ such that the following holds.

•

For any $\delta>0$ , there exists $M_{-,\delta}>0$ such that

$f\leq M_{-,\delta}(|z|^{b_{-}}+1)$

on $\bigl{\{}z\in D_{a}(R)\,\big{|}-\pi/2a<\arg(z)<-\delta\bigr{\}}$ .
•

For any $\delta>0$ , there exists $M_{+,\delta}>0$ such that

$f\leq M_{+,\delta}(|z|^{b_{+}}+1)$

on $\bigl{\{}z\in D_{a}(R)\,\big{|}\,\delta<\arg(z)<\pi/2a\bigr{\}}$ .
•

There exist an exhaustive family $\{K_{i}\}$ of $D_{a}(R)$ and $C>0$ such that

$f\leq C(|z|^{\kappa}+1)$

on $\partial K_{i}$ .

Then, there exists $C_{1}>0$ such that $f\leq C_{1}(|z|^{b}+1)$ on $D_{a}(R)$ .

Proof We explain the case $0\leq b_{-}\leq b_{+}$ . The other case can be discussed similarly. For any $a_{1}>a$ , we take a decreasing sequence $c_{i}$ in $\{a_{1}<c<2a_{1}\}$ such that $\lim c_{i}=a_{1}$ . We set $K_{1,i}=K_{i}\cap D_{c_{i}}(R)$ . Then, $\{K_{1,i}\}$ is an exhaustive family of $D_{a_{1}}(R)$ , and there exists $C_{2}>0$ such that $f\leq C_{2}(|z|^{\kappa}+1)$ on $\partial K_{1,i}$ . Hence, by making $a$ larger, we may assume that $\kappa<a$ from the beginning.

There exists $C_{3}>0$ such that $f-C_{3}\mathop{\rm Re}\nolimits\bigl{(}z^{b_{+}}+1\bigr{)}\leq 0$ on $\bigl{\{}z\in D_{a}(R)\,\big{|}\,\pi/4a<|\arg(z)|<\pi/2a\bigr{\}}$ . Then, we obtain the claim of the corollary from Proposition 4.1. ∎

4.1.2. Nevanlinna formula

We set $\mathbb{H}:=\{z\in{\mathbb{C}}\,|\,\mathop{\rm Im}\nolimits(z)>0\}$ . For any $R\geq 0$ , we set $\mathbb{H}(R):=\bigl{\{}z\in\mathbb{H}\,\big{|}\,|z|>R\bigr{\}}$ . Let $\overline{\mathbb{H}}$ and $\overline{\mathbb{H}}(R)$ denote the closure of $\mathbb{H}$ and $\overline{\mathbb{H}}(R)$ , respectively. Let $(x,y)$ be the real coordinate system on ${\mathbb{C}}$ determined by $z=x+\sqrt{-1}y$ .

Let $f$ be an ${\mathbb{R}}$ -valued $C^{\infty}$ -function on $\overline{\mathbb{H}}(R)$ for some $R\geq 0$ . Suppose that there exist $C>0$ , $0<\rho<1$ and $\delta\geq 0$ such that the following conditions are satisfied.

•

$f$ is $C^{\infty}$ in $\mathbb{H}(R)$ , and $|\triangle(f)|\leq C(1+y^{2})^{-1-\delta}$ . Here, $\triangle=\partial_{x}^{2}+\partial_{y}^{2}$ .
•

There exists an exhaustive family $\{K_{i}\}$ of $\mathbb{H}(R)$ such that $|f|\leq y+C(|z|^{\rho}+1)$ on $\partial K_{i}$ $(i=1,2,\ldots)$ .

Proposition 4.4.

There exist $-1\leq a(f)\leq 1$ and a constant $C_{1}>0$ such that the following holds on $\mathbb{H}(R)$ :

|f-a(f)y|\leq C_{1}(|z|^{\rho}+1).

If $|f|$ is bounded on $\partial\mathbb{H}(R)$ , we obtain the following stronger estimate on $\mathbb{H}(R)$ for a positive constant $C_{2}>0$ :

|f-a(f)y|\leq C_{2}\log(2+y).

If $|f|$ is bounded on $\partial\mathbb{H}(R)$ , and if $\delta>0$ , then $|f-a(f)y|$ is bounded on $\mathbb{H}(R)$ .

Proof There exists a $C^{\infty}$ -function $\widetilde{f}$ on $\overline{\mathbb{H}}$ such that $\widetilde{f}=f$ on $\overline{\mathbb{H}}(R+1)$ . There exists $\widetilde{C}>0$ such that the following holds.

•

$|\triangle\widetilde{f}|\leq\widetilde{C}(1+y^{2})^{-1-\delta}$ .
•

There exists an exhaustive family $\{\widetilde{K}_{i}\}$ such that $|\widetilde{f}|\leq y+\widetilde{C}(1+|z|^{\rho})$ .

If $|f|$ is bounded on $\partial\mathbb{H}(R)$ , then $|\widetilde{f}|$ is bounded on $\partial\mathbb{H}$ . It is enough to obtain the estimate for $\widetilde{f}$ . Therefore, we may assume $R=0$ from the beginning.

Let $\zeta$ be a complex variable, and let $(\xi,\eta)$ be the real variables determined by $\zeta=\xi+\sqrt{-1}\eta$ . On $\{y>0\}$ , we consider the following function

F_{1}(z):=\frac{1}{4\pi}\int_{-\infty}^{\infty}d\xi\int_{0}^{\infty}\,d\eta\left(\triangle f(\zeta)\cdot\log\left(\frac{(x-\xi)^{2}+(y-\eta)^{2}}{(x-\xi)^{2}+(y+\eta)^{2}}\right)\right).

By Lemma 4.5 below, the integral is convergent, and we obtain the estimate

F_{1}(z)=\left\{\begin{array}[]{ll}O\bigl{(}\log(1+y)\bigr{)}&(\delta=0)\\ \\ O\bigl{(}y(1+y)^{-1}\bigr{)}&(\delta>0)\end{array}\right.

Moreover, we obtain $\triangle F_{1}=\triangle f$ , and hence $F_{2}:=f-F_{1}$ is a harmonic function on $\mathbb{H}$ , and continuous on $\overline{\mathbb{H}}$ .

We set $\phi:=\mathop{\rm Re}\nolimits\bigl{(}e^{-\pi\rho\sqrt{-1}/2}(z+\sqrt{-1})^{\rho}\bigr{)}$ , where we consider the branch of $(z+\sqrt{-1})^{\rho}$ such that $(b\sqrt{-1}+\sqrt{-1})^{\rho}=(b+1)^{\rho}e^{\pi\rho\sqrt{-1}/2}$ for $b>0$ . It is a harmonic function, and there exists $\epsilon>0$ such that $\epsilon(|z|^{\rho}+1)\leq\phi$ on $\overline{\mathbb{H}}$ . By the assumption, there exist $C_{10}>0$ such that $|F_{2}|\leq C_{10}\bigl{(}y+\phi\bigr{)}$ on $\partial K_{i}$ $(i=1,2,\ldots)$ . Because $F_{2}$ is harmonic, we obtain $|F_{2}|\leq C_{10}\bigl{(}y+\phi\bigr{)}$ on $K_{i}$ , and hence on $\overline{\mathbb{H}}$ . In particular, we obtain $|F_{2}|\leq C_{10}\phi$ on $\partial\mathbb{H}$ , and $F_{2}+C_{10}\bigl{(}y+\phi\bigr{)}\geq 0$ on $\mathbb{H}$ . Then, according to the Nevanlinna formula [22, §14, Theorem 1] (see also [22, Remark in page 101]), there exists a real number $c$ such that the following holds for $y>0$ :

F_{2}+C_{10}\bigl{(}y+\phi\bigr{)}=cy+\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{F_{2}(t,0)+C_{10}\phi(t,0)\,dt}{(t-x)^{2}+y^{2}}.

Note that

\phi(x,y)=\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{\phi(t,0)}{(t-x)^{2}+y^{2}}\,dt.

We obtain

f=(c-C_{10})y+F_{1}+\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{f(t,0)\,dt}{(t-x)^{2}+y^{2}}.

Note that $|f|=|F_{2}|\leq C_{10}\phi$ on $\partial\mathbb{H}$ . We obtain that $\Bigl{|}\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{f(t,0)\,dt}{(t-x)^{2}+y^{2}}\Bigr{|}\leq C_{10}\phi$ on $\mathbb{H}$ . Then, we can easily obtain the claims of the proposition. ∎

In the proof, we have used the following lemma.

Lemma 4.5.

There exists a constant $C_{0}>0$ such that the following holds for any $z\in\mathbb{H}$ :

0\leq\int_{-\infty}^{\infty}\,d\xi\int_{0}^{\infty}\frac{d\eta}{1+\eta^{2}}\left(\log\left|\frac{\zeta-\overline{z}}{\zeta-z}\right|^{2}\right)\leq C_{0}\log(1+y).

(41)

For any $\delta>0$ , there exists a constant $C_{\delta}>0$ such that the following holds for any $z$ with $y>0$ :

0\leq\int_{-\infty}^{\infty}\,d\xi\int_{0}^{\infty}\frac{d\eta}{(1+\eta^{2})^{1+\delta}}\left(\log\left|\frac{\zeta-\overline{z}}{\zeta-z}\right|^{2}\right)\leq C_{\delta}\frac{y}{1+y}.

(42)

Proof Let us study the case $\delta=0$ . We set $Z_{1}:=\{(\xi,\eta)\,|\,\eta\geq 2y\}$ , $Z_{2}:=\{(\xi,\eta)\,|\,y\leq 2\eta\leq 4y\}$ , and $Z_{3}:=\{(\xi,\eta)\,|\,0<2\eta\leq y\}$ . We obtain

\int_{Z_{1}}\frac{d\xi d\eta}{1+\eta^{2}}\log\left|\frac{\zeta-\overline{z}}{\zeta-z}\right|^{2}=\int_{Z_{1}}\frac{d\xi d\eta}{1+\eta^{2}}\log\left(\frac{\xi^{2}+(\eta+y)^{2}}{\xi^{2}+(\eta-y)^{2}}\right)\\ =\int_{Z_{1}}\frac{d\xi\,d\eta}{1+\eta^{2}}\log\left(1+\frac{4y\eta}{\xi^{2}+(\eta-y)^{2}}\right)\\ \leq\int_{-\infty}^{\infty}\,d\xi\int_{2}^{\infty}\frac{y^{2}}{1+y^{2}\eta^{2}}\frac{4\eta d\eta}{\bigl{(}\xi^{2}+(\eta-1)^{2}\bigr{)}}.

(43)

For $y\geq 1$ , (43) is dominated by

\int_{-\infty}^{\infty}\,d\xi\int_{2}^{\infty}\frac{4d\eta}{\eta\bigl{(}\xi^{2}+(\eta-1)^{2}\bigr{)}}<\infty.

For $0<y\leq 1$ , (43) is dominated by

C_{1}\int_{2}^{\infty}\frac{y^{2}\,d\eta}{1+y^{2}\eta^{2}}\leq C_{1}\int_{-\infty}^{\infty}\frac{y^{2}\,d\eta}{1+y^{2}\eta^{2}}\leq C_{2}y.

Similarly, we obtain

\int_{Z_{2}}\frac{d\xi\,d\eta}{1+\eta^{2}}\log\left|\frac{\zeta-\overline{z}}{\zeta-z}\right|^{2}=\int_{Z_{2}}\frac{d\xi\,d\eta}{1+\eta^{2}}\log\left(1+\frac{4y\eta}{\xi^{2}+(y-\eta)^{2}}\right)\\ \leq\int_{|\xi|\geq 1}d\xi\int_{\frac{1}{2}}^{2}\frac{y^{2}}{1+y^{2}\eta^{2}}\frac{4\eta d\eta}{\bigl{(}\xi^{2}+(\eta-1)^{2}\bigr{)}}\\ +\int_{|\xi\leq 1}d\xi\int_{\frac{1}{2}}^{2}\frac{y^{2}\,d\eta}{1+y^{2}\eta^{2}}\log\left(1+\frac{\eta}{\xi^{2}+(\eta-1)^{2}}\right).

(44)

If $y\geq 1$ , (44) is dominated by

\int_{|\xi|\geq 1}d\xi\int_{\frac{1}{2}}^{2}\frac{4d\eta}{\eta\bigl{(}\xi^{2}+(\eta-1)^{2}\bigr{)}}+\\ \int_{|\xi\leq 1}d\xi\int_{\frac{1}{2}}^{2}\frac{d\eta}{\eta^{2}}\log\left(1+\frac{\eta}{\xi^{2}+(\eta-1)^{2}}\right)<\infty

(45)

If $y\leq 1$ , (44) is dominated by

8y^{2}\int_{|\xi|\geq 1}\frac{d\xi}{\xi^{2}}\int_{\frac{1}{2}}^{2}d\eta+y^{2}\int_{|\xi|\leq 1}d\xi\int_{\frac{1}{2}}^{2}d\eta\log\left(1+\frac{\eta}{\xi^{2}+(\eta-1)^{2}}\right)\leq C_{3}y^{2}.

As for the integral over $Z_{3}$ , we obtain

\int_{Z_{3}}\frac{d\xi\,d\eta}{1+\eta^{2}}\log\left|\frac{\zeta-\overline{z}}{\zeta-z}\right|^{2}\leq\int_{Z_{3}}\frac{d\xi\,d\eta}{1+\eta^{2}}\frac{4y\eta}{\xi^{2}+(\eta-y)^{2}}\\ \leq C_{10}\int_{0}^{y/2}\frac{d\eta}{1+\eta^{2}}\frac{y\eta}{y-\eta}\leq 2C_{10}\int_{0}^{y/2}\frac{\eta d\eta}{1+\eta^{2}}\leq C_{11}\log(1+y).

(46)

Thus, we obtain (41) in the case $\delta=0$ .

As for the case $\delta>0$ , the integrals over $Z_{i}$ $(i=1,2)$ are dominated as in the case of (41). The integral over $Z_{3}$ is dominated by $\int_{0}^{y/2}(1+\eta^{2})^{-1-\delta}\,\eta\,d\eta<C_{20}(\delta)y(1+y)^{-1}$ . Thus, we obtain (42). ∎

4.1.3. Holomorphic line bundles with a Hermitian metric on upper half plane

We state a consequence of the Nevanlinna formula (Proposition 4.4) on a holomorphic line bundle $L$ with a Hermitian metric $h$ on $\mathbb{H}$ , which is fundamental in the classification of solutions of Toda equations in terms of parabolic structures. Let $R(h)$ denote the curvature of the Chern connection associated with $(L,h)$ . By using the standard Euclidean metric $g=dz\,d\overline{z}$ , we assume the following condition.

•

$|R(h)|_{h,g}=O\bigl{(}(1+y^{2})^{-1}\bigr{)}$ .

Note that this condition is analogue to the acceptability condition in [38, 39].

Lemma 4.6.

Let $v$ be a global frame of $L$ . If there exist $C>0$ , $0<\rho<1$ , and an exhaustive family $\{K_{i}\}$ of $\mathbb{H}$ such that $\bigl{|}\log|v|_{h}\bigr{|}\leq y+C(|z|^{\rho}+1)$ on $\partial K_{i}$ , then there exists $-1\leq a(h,v)\leq 1$ such that $\log|v|_{h}-a(h,v)y=O(|z|^{\rho}+1)$ on $\mathbb{H}$ . If $\log|v|_{h}$ is bounded on $\partial\mathbb{H}$ , then we obtain $\log|v|_{h}-a(h,v)y=O\bigl{(}\log(y+2)\bigr{)}$ on $\mathbb{H}$ .

Proof Because $R(h)=\frac{1}{4}\triangle(\log|v|^{2}_{h})\,d\overline{z}\,dz$ , we obtain $\triangle(\log|v|_{h}^{2})=O\bigl{(}(1+y^{2})^{-1}\bigr{)}$ . Then, the claim follows from Proposition 4.4. ∎

Definition 4.7.

The number $a(h,v)$ is called the parabolic order of $v$ with respect to $h$ .

4.2. Finite exponential sums and their perturbation

Let us recall some results in [41, 48], which are summarized in [21].

4.2.1. Finite exponential sums

Let $c_{0}<c_{1}<c_{2}<\cdots<c_{n}$ be real numbers. Let $a_{i}$ $(i=0,\ldots,n)$ be non-zero complex numbers. We consider the entire function

F(\zeta)=\sum_{i=0}^{n}a_{i}e^{\sqrt{-1}c_{i}\zeta}.

We set $Z(F):=\{\zeta\in{\mathbb{C}}\,|\,F(\zeta)=0\}$ . It is easy to see that there exists $L>0$ such that any $\zeta\in Z(F)$ satisfies $|\mathop{\rm Im}\nolimits(\zeta)|<L$ . For any $x_{1}<x_{2}$ , we set $Y(x_{1},x_{2}):=\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,|\mathop{\rm Im}\nolimits(\zeta)|\leq L,\,\,x_{1}\leq\mathop{\rm Re}\nolimits(\zeta)\leq x_{2}\bigr{\}}$ .

Proposition 4.8 ([21, 41, 48]).

If $Z(F)\cap\bigcup\{\mathop{\rm Re}\nolimits(\zeta)=x_{i}\}=\emptyset$ , we obtain

-n+\frac{c_{n}-c_{0}}{2\pi}(x_{2}-x_{1})\leq\#\Bigl{(}Y(x_{1},x_{2})\cap Z(F)\Bigr{)}\leq n+\frac{c_{n}-c_{0}}{2\pi}(x_{2}-x_{1}).

It particularly implies that the order of the zero of $F$ at any point is not larger than $n$ . ∎

Note that the proposition implies that $Z(F)\cap\{\mathop{\rm Re}\nolimits(\zeta)=x_{i}\}\leq n$ . Hence, we obtain the following for any $x_{1}<x_{2}$ :

-3n+\frac{c_{n}-c_{0}}{2\pi}(x_{2}-x_{1})\leq\#\Bigl{(}Y(x_{1},x_{2})\cap Z(F)\Bigr{)}\leq 3n+\frac{c_{n}-c_{0}}{2\pi}(x_{2}-x_{1}).

4.2.2. Perturbation

Let $G$ be a holomorphic function defined on $W(C_{1},C_{2}):=\{\zeta\in{\mathbb{C}}\,|\,\mathop{\rm Re}\nolimits(\zeta)>C_{1},\,\,|\mathop{\rm Im}\nolimits(\zeta)|<C_{2}\}$ such that

\lim_{\mathop{\rm Re}\nolimits(\zeta)\to\infty}|F(\zeta)-G(\zeta)|=0.

We set $Z(G):=\{\zeta\in W(C_{1},C_{2})\,|\,G(\zeta)=0\}$ . For any closed subset $K\subset{\mathbb{C}}$ and $\zeta\in{\mathbb{C}}$ , let $d(\zeta,K)$ denote the Euclidean distance between $\zeta$ and $A$ , i.e., $d(\zeta,A)=\min\bigl{\{}|\zeta^{\prime}-\zeta|\,\big{|}\,\zeta^{\prime}\in A\bigr{\}}$ .

Proposition 4.9.

Take $0<C_{2}^{\prime}<C_{2}$ .

•

There exist $C_{3}>0$ such that the following holds for any $C_{1}+1<x_{1}<x_{2}$ :

$\#\Bigl{(}Y(x_{1},x_{2})\cap W(C_{1},C_{2}^{\prime})\cap Z(G)\Bigr{)}\leq C_{3}(1+|x_{2}-x_{1}|).$
•

There exist $C_{1}^{\prime}>C_{1}$ and $C_{4}>0$ such that $|G(\zeta)|\geq C_{4}\delta^{n}$ holds for any $0<\delta\leq 1$ and any $\zeta\in W(C_{1}^{\prime},C_{2}^{\prime})$ satisfying $d(\zeta,Z(G))\geq\delta$ .

Proof We apply the argument in [48]. Let us prove the first claim in the case $|x_{2}-x_{1}|=1$ . Assume that there exists a sequence $x_{j}$ such that

\#\Bigl{(}Y(x_{j},x_{j}+1)\cap W(C_{1},C_{2}^{\prime})\cap Z(G)\Bigr{)}\to\infty.

Let

\Psi_{j}:\{\zeta\in{\mathbb{C}}\,|\,-\epsilon<\mathop{\rm Re}\nolimits(\zeta)<1+\epsilon,\,|\mathop{\rm Im}\nolimits(\zeta)|<C_{2}\}\simeq\\ \{\zeta\in{\mathbb{C}}\,|\,x_{j}-\epsilon<\mathop{\rm Re}\nolimits(\zeta)<x_{j}+1+\epsilon,|\mathop{\rm Im}\nolimits(\zeta)|<C_{2}\}

(47)

be the isomorphism defined by $\Psi_{j}(\zeta)=\zeta+x_{j}$ . Going into a subsequence, we may assume that the sequence $\Psi_{j}^{\ast}(F)$ is convergent to $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ , where $b_{i}$ are complex numbers such that $|b_{i}|=|a_{i}|$ . It implies that the sequence $\Psi_{j}^{\ast}(G)$ is convergent to $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ . Then, we obtain the contradiction by Proposition 4.8 and Lemma 4.12 below. Hence, there exists a constant $C_{10}>0$ such that

\#\Bigl{(}Y(x,x+1)\cap W(C_{1},C_{2}^{\prime})\cap Z(G)\Bigr{)}<C_{10}

for any $x>C_{1}+1$ .

Take any $C_{1}+1<x_{1}<x_{2}$ . We set $m_{0}:=\min\{m\in{\mathbb{Z}}\,|\,x_{2}-x_{1}\leq m\}\geq 1$ . Then, we obtain

\#\Bigl{(}Y(x_{1},x_{2})\cap W(C_{1},C_{2}^{\prime})\cap Z(G)\Bigr{)}\leq\\ \#\Bigl{(}Y(x_{1},x_{1}+m_{0})\cap W(C_{1},C_{2}^{\prime})\cap Z(G)\Bigr{)}\leq m_{0}C_{10}\\ \leq(1+|x_{2}-x_{0}|)C_{10}.

(48)

Thus, we obtain the first claim.

To prove the second claim, we prepare the following lemma.

Lemma 4.10.

There exists $C_{1}^{\prime\prime}>C_{1}$ such that the order of the zero of $G$ at any point of $\overline{W(C_{1}^{\prime\prime},C^{\prime}_{2})}\cap Z(G)$ is not larger than $n$ .

Proof Suppose the contrary. Let $\zeta_{j}\in\overline{W(C_{1},C_{2}^{\prime})}\cap Z(G)$ such that the order of zero of $G$ at $\zeta_{j}$ is strictly larger than $n$ . Let

\Psi_{j}:\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,-1<\mathop{\rm Re}\nolimits(\zeta)<1,\,|\mathop{\rm Im}\nolimits(\zeta)|<C_{2}\bigr{\}}\simeq\\ \bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,-1<\mathop{\rm Re}\nolimits(\zeta)-\mathop{\rm Re}\nolimits(\zeta_{j})<1,\,|\mathop{\rm Im}\nolimits(\zeta)|<C_{2}\bigr{\}}

(49)

be the isomorphism determined by $\Psi_{j}(\zeta)=\zeta+\mathop{\rm Re}\nolimits(\zeta_{j})$ . We may assume that the sequence $\Psi_{j}^{\ast}(F)$ is convergent to a holomorphic function of the form $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ , where $b_{i}$ are complex numbers such that $|b_{i}|=|a_{i}|$ . It particularly implies that $\Psi_{j}^{\ast}(G)$ is convergent to $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ . We may also assume that $\sqrt{-1}\mathop{\rm Im}\nolimits(\zeta_{j})$ is convergent to $\sqrt{-1}\beta$ for some $-C_{2}^{\prime}\leq\beta\leq C_{2}^{\prime}$ . Because the order of zero of $\Psi_{j}^{\ast}(G)$ at $\sqrt{-1}\mathop{\rm Im}\nolimits(\zeta_{j})$ are strictly larger than $n$ , we obtain that the order of $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ is strictly larger than $n$ . But, it contradicts Proposition 4.8. ∎

Let us prove the second claim. Suppose the contrary. There exist sequences $\epsilon_{j}>0$ , $0<\delta_{j}\leq 1$ and $\zeta_{j}\in W(C_{1}^{\prime\prime},C_{2}^{\prime})$ satisfying $\lim\epsilon_{j}=0$ , $d(\zeta_{j},Z(G))\geq\delta_{j}$ and $|G(\zeta)|\leq\epsilon_{j}\delta_{j}^{n}$ . We shall deduce a contradiction.

Let us consider the case that the sequence $\zeta_{j}$ contains a bounded sequence. By going to a subsequence, the sequence $\zeta_{j}$ is convergent to $\zeta_{\infty}\in\overline{W(C_{1}^{\prime\prime},C_{2}^{\prime})}$ . It is easy to observe that $\zeta_{\infty}\in Z(G)$ , and the order of zero at $\zeta_{\infty}$ is strictly larger than $n$ . Hence, we obtain a contradiction by Lemma 4.10.

Let us consider the case that the sequence $\zeta_{j}$ does not contain a bounded sequence. In particular, $\mathop{\rm Re}\nolimits(\zeta_{j})\to\infty$ as $j\to\infty$ . We consider the maps $\Psi_{j}$ as in (49). We may assume that the sequence $\Psi_{j}^{\ast}(F)$ is convergent to a holomorphic function of the form $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ , where $b_{i}$ are complex numbers such that $|b_{i}|=|a_{i}|$ . It particularly implies that $\Psi_{j}^{\ast}(G)$ is convergent to $\sum b_{i}e^{\sqrt{-1}c_{i}\zeta}$ . Then, we obtain a contradiction by Proposition 4.8 and Lemma 4.13 below. Thus, we obtain the second claim of the proposition. ∎

Corollary 4.11.

For any $0<\delta\leq 1$ , there exists $C_{5}>0$ such that we obtain $|G(\zeta)|\geq C_{4}\delta^{n}$ for any $\zeta\in W(C_{1},C_{2}^{\prime})$ satisfying $d(\zeta,Z(G))\geq\delta$ . ∎

4.2.3. Appendix

Let $Y\subset{\mathbb{C}}$ be a connected open subset. Let $Y_{1}$ be a relatively compact open subset of $Y$ . Let $F_{i}:Y\longrightarrow{\mathbb{C}}$ $(i=1,2,\ldots)$ be holomorphic functions which uniformly converge to a holomorphic function $F_{\infty}:Y\longrightarrow{\mathbb{C}}$ . Note that any derivatives $\partial_{z}^{j}F_{i}$ converges to $\partial_{z}^{j}F_{\infty}$ . Assume that $F_{\infty}$ is not constantly $0$ .

For $i=1,2,\ldots,\infty$ , we set $Z(F_{i}):=\{\zeta\in Y\,|\,F_{i}(\zeta)=0\}$ . Let $\mathfrak{m}_{i}(\zeta)\in{\mathbb{Z}}_{>0}$ denote the order of the zero of $F_{i}$ at $\zeta$ .

Lemma 4.12.

There exists $i_{0}$ such that the following holds for any $i\geq i_{0}$ :

\sum_{\zeta\in Z(F_{i})\cap\overline{Y}_{1}}\mathfrak{m}_{i}(\zeta)\leq\sum_{\zeta\in Z(F_{\infty})\cap\overline{Y}_{1}}\mathfrak{m}_{\infty}(\zeta).

Proof We may assume that $Y_{1}$ is simply connected. There exists a relatively compact open subset $Y_{2}\subset Y$ such that (i) $\partial Y_{2}$ is smooth, (ii) $Y_{1}\subset Y_{2}$ , (iii) $Z(F_{\infty})\cap\overline{Y}_{1}=Z(F_{\infty})\cap\overline{Y}_{2}$ . The condition (iii) implies that $F_{\infty}(\zeta)\neq 0$ for any $\zeta\in\partial Y_{2}$ . There exists $\epsilon>0$ such that $\min_{\zeta\in\partial Y_{2}}|F_{\infty}(\zeta)|>2\epsilon$ . There exists $i_{0}$ such that $|F_{i}(\zeta)-F_{\infty}(\zeta)|\leq\epsilon/2$ for any $\zeta\in\partial Y_{2}$ and $i\geq i_{0}$ . Then, the claim holds according to Rouché’s theorem. ∎

Let $n$ be a positive integer such that $\mathfrak{m}_{\infty}(\zeta)\leq n$ for any $\zeta\in\overline{Y}_{1}$ . For any closed subset $K\subset{\mathbb{C}}$ , let $d(\zeta,K)$ denote the Euclidean distance between $\zeta$ and $K$ . There exists $\delta_{0}>0$ such that $\{\zeta\in{\mathbb{C}}\,|\,d(\zeta,\overline{Y}_{1})\leq 2\delta_{0}\}\subset Y$ . For any $0<\delta<\delta_{0}$ , we set $W_{i}(\delta):=\{\zeta\in Y_{1}\,|\,d(\zeta,Z(F_{i}))\geq\delta\}$ .

Lemma 4.13.

There exist $C>0$ and $i_{0}$ such that we obtain $|F_{i}|\geq C\delta^{n}$ on $W_{i}(\delta)$ for any $i\geq i_{0}$ and any $0<\delta<\delta_{0}$ .

Proof Assume the contrary. Then, there exist a sequence of positive numbers $C_{1}>C_{2}>\cdots$ , a sequence $i(j)$ , positive numbers $\delta_{j}>0$ and $\zeta_{j}\in W_{i(j)}(\delta_{j})$ such that $\lim C_{j}=0$ and that $|F_{i(j)}(\zeta_{j})|<C_{j}\delta_{j}^{n}$ . Going to a subsequence, we may assume that $i(j)=j$ . We may assume $\delta_{j}=d(\zeta_{j},Z(F_{j}))$ . Going to a subsequence, we may assume that the sequence $\zeta_{j}$ is convergent to $\zeta_{\infty}\in\overline{Y}_{1}$ . Because $F_{\infty}(\zeta_{\infty})=\lim F_{j}(\zeta_{j})$ , we obtain $F_{\infty}(\zeta_{\infty})=0$ , and hence $\zeta_{\infty}\in Z(F_{\infty})\cap\overline{Y}_{1}$ . For any $\epsilon>0$ , there exists $j_{0}$ such that $d(\zeta_{\infty},Z(F_{j}))<\epsilon$ for any $j\geq j_{0}$ . Hence, we obtain $\lim\delta_{j}=0$ . Going to a subsequence, we may assume that the sequence $\delta_{j}$ is decreasing.

Fix a relatively compact open neighbourhood $U$ of $\zeta_{\infty}$ in $Y$ such that $\overline{U}\cap Z(F_{\infty})=\{\zeta_{\infty}\}$ . We set $\ell:=\mathfrak{m}_{\infty}(\zeta_{\infty})\leq n$ . Going to a subsequence, we may assume that $F_{j}$ is nowhere vanishing on $\partial U$ for any $j$ . We set $S_{j}:=U\cap Z(F_{j})$ . We obtain $\sum_{a\in S_{j}}\mathfrak{m}_{j}(a)=\ell$ . We set

G_{U,j}(\zeta):=F_{j|U}(\zeta)\cdot\prod_{a\in S_{j}}(\zeta-a)^{-\mathfrak{m}_{j}(a)},

which is nowhere vanishing on $U$ for any $j$ . Because the sequence $F_{j}$ is convergent to $F_{\infty}$ , the sequence $\prod_{a\in S_{j}}(\zeta-a)^{\mathfrak{m}_{j}(a)}$ is convergent to $(\zeta-\zeta_{\infty})^{\ell}$ . The sequence $G_{U,j}$ is convergent to $G_{U,\infty}(\zeta):=F_{\infty|U}(\zeta)(\zeta-\zeta_{\infty})^{-\ell}$ , which is also nowhere vanishing on $U$ .

By the assumption, $|\zeta_{j}-a|\geq\delta_{j}$ for any $a\in S_{j}$ . Hence, $\prod_{a\in S_{j}}(\zeta_{j}-a)^{\mathfrak{m}_{j}(a)}\geq\delta_{j}^{\ell}$ . Then, there exists $A>0$ such that the following holds for any $j$ :

|F_{j}(\zeta_{j})|\geq A\cdot\delta_{j}^{\ell}.

Hence, we obtain $A\delta_{j}^{\ell}\leq|F_{j}(\zeta_{j})|\leq C_{j}\delta_{j}^{n}$ . It implies $0<A\leq C_{j}\delta_{j}^{n-\ell}\to 0$ , which is a contradiction. ∎

4.3. Holomorphic functions with multiple growth orders

4.3.1. Growth orders

Let $\varpi:\widetilde{{\mathbb{C}}}\longrightarrow{\mathbb{C}}$ be the oriented real blowing up at $0$ . We set ${\mathbb{C}}^{\ast}:={\mathbb{C}}\setminus\{0\}$ . Let $\iota:{\mathbb{C}}^{\ast}\longrightarrow\widetilde{{\mathbb{C}}}$ denote the natural inclusion. Let $z$ denote the standard coordinate of ${\mathbb{C}}$ . We identify $\widetilde{{\mathbb{C}}}$ with ${\mathbb{R}}_{\geq 0}\times S^{1}$ by the polar decomposition of $z$ .

Notation 4.14.

Let $Q=(0,e^{\sqrt{-1}\theta_{0}})\in\varpi^{-1}(0)$ . Let $\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ denote the stalk of the sheaf $\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})$ at $Q$ . We fix a branch of $\log z$ around $Q$ .

•

Let $\mathfrak{I}(Q)\subset\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ be the ${\mathbb{C}}$ -linear subspace generated by $z^{-a}$ $(a>0)$ , i.e., $\mathfrak{I}(Q)=\bigoplus_{a>0}{\mathbb{C}}\,z^{-a}$ .
•

Note that for any $\mathfrak{a}\in\mathfrak{I}(Q)\setminus\{0\}$ , $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})$ induces a continuous function on a neighbourhood of $Q$ . The continuous function is also denoted by $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})$ .
•

For any $\mathfrak{a}\in\mathfrak{I}(Q)\neq 0$ , we have the expression $\mathfrak{a}=\alpha z^{-\rho}+\sum_{0<c<\rho}\alpha_{c}z^{-c}$ where $\alpha\neq 0$ . We set $\deg(\mathfrak{a}):=\rho$ .
•

For $\mathfrak{a},\mathfrak{b}\in\mathfrak{I}(Q)$ , we say $\mathfrak{a}\prec_{Q}\mathfrak{b}$ if $|\mathfrak{b}-\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a})(Q)>0$ or $\mathfrak{a}=\mathfrak{b}$ . It defines a partial order $\prec_{Q}$ on $\mathfrak{I}(Q)$ .

Remark 4.15.

Note that the orders $\prec_{Q}$ are opposite to the order used in [29].

4.3.2. Regularly bounded holomorphic functions

Definition 4.16.

We say that $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ is regularly bounded if there exist a neighbourhood $\mathcal{U}$ of $Q$ in $\widetilde{{\mathbb{C}}}$ , non-zero complex numbers $\alpha_{1},\ldots,\alpha_{m}$ , and mutually distinct real numbers $c_{1},c_{2},\ldots,c_{m}$ such that the following holds.

•

$f$ induces a holomorphic function on $\mathcal{U}\setminus\varpi^{-1}(0)$ .
•

$|f(z)-\sum_{i=1}^{m}\alpha_{i}z^{\sqrt{-1}c_{i}}|\to 0$ as $|z|\to 0$ in $\mathcal{U}\setminus\varpi^{-1}(0)$ .

4.3.3. Holomorphic functions with single growth order

Definition 4.17.

We say that $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has a single growth order if there exist $\mathfrak{a}(f,Q)\in\mathfrak{I}(Q)$ , $a(f,Q)\in{\mathbb{R}}$ , $j(f,Q)\in{\mathbb{Z}}_{\geq 0}$ such that $e^{-\mathfrak{a}(f,Q)}z^{-a(f,Q)}(\log z)^{-j(f,Q)}f$ is regularly bounded.

Definition 4.18.

Suppose that $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has single growth order at $Q$ .

•

We say that $f$ is simply positive (resp. negative) at $Q$ if

|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))(Q)>0\quad\mbox{(resp. $|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))(Q)<0$)}.

•

If $\mathfrak{a}(f,Q)=0$ , then we say that $f$ is neutral at $Q$ .
•

If $\mathfrak{a}(f,Q)\neq 0$ but $|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))(Q)=0$ , we say that $f$ is turning at $Q$ .

The following lemma is easy to see.

Lemma 4.19.

If $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has a single growth order, there exists a neighbourhood $\mathcal{U}$ of $Q$ such that (i) $\mathcal{U}\cap\varpi^{-1}(0)$ is connected, (ii) $f$ induces a holomorphic function on $\mathcal{U}\setminus\varpi^{-1}(0)$ , (iii) $f$ has a single growth order at any point $Q^{\prime}\in\mathcal{U}^{\prime}\cap\varpi^{-1}(0)$ , (iv) $\mathfrak{a}(f,Q^{\prime})=\mathfrak{a}(f,Q)$ . ∎

Note that (iv) implies that the conditions “simply positive”, “simply negative” and “neutral” are preserved if $\mathcal{U}$ is sufficiently small. If $f$ is turning at $Q$ , $|\mathfrak{a}(f,Q)|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q))$ is positive on a connected component of $(\mathcal{U}\cap\varpi^{-1}(0))\setminus\{Q\}$ and negative on the other connected component of $(\mathcal{U}\cap\varpi^{-1}(0))\setminus\{Q\}$ .

Lemma 4.20.

Let $f$ be a section of $\iota_{\ast}\mathcal{O}_{{\mathbb{C}}^{\ast}}$ on an open subset $V\subset\widetilde{{\mathbb{C}}}$ . Let $I\subset V\cap\varpi^{-1}(0)$ be an interval. If $f$ has single growth order at each $Q\in I$ , then we obtain $\mathfrak{a}(f,Q_{1})=\mathfrak{a}(f,Q_{2})$ for any $Q_{1},Q_{2}\in I$ .

Proof It follows from Lemma 4.19. ∎

4.3.4. Holomorphic functions with multiple growth orders

Definition 4.21.

We say that $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has multiple growth orders if $f$ is expressed as a finite sum $\sum_{i=1}^{N}f_{i}$ such that (i) each $f_{i}$ has a single growth order, (ii) $\mathfrak{a}(f_{i},Q)\neq\mathfrak{a}(f_{j},Q)$ $(i\neq j)$ .

Lemma 4.22.

Suppose that $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has multiple growth orders.

•

There exists a finite subset $\mathcal{I}\subset\mathfrak{I}(Q)$ and an expression $f=\sum_{\mathfrak{a}\in\mathcal{I}}f_{\mathfrak{a}}$ such that (i) for two distinct elements $\mathfrak{a},\mathfrak{b}\in\mathcal{I}$ , neither $\mathfrak{a}\prec_{Q}\mathfrak{b}$ nor $\mathfrak{b}\prec_{Q}\mathfrak{a}$ holds, (ii) each $f_{\mathfrak{a}}$ has single growth order with $\mathfrak{a}(f_{\mathfrak{a}},Q)=\mathfrak{a}$ . Such an expression is called reduced.
•

There exists a neighbourhood $\mathcal{U}$ of $Q$ such that (i) $\mathcal{U}\cap\varpi^{-1}(0)$ is connected, (ii) $f$ induces a holomorphic function on $\mathcal{U}\setminus\varpi^{-1}(0)$ , (iii) $f$ has single growth order at any point $Q^{\prime}\in\bigl{(}\mathcal{U}\cap\varpi^{-1}(0)\bigr{)}\setminus\{Q\}$ , and $\mathfrak{a}(f,Q^{\prime})$ is the unique maximal element of the partially ordered set $(\mathcal{I},\prec_{Q^{\prime}})$ .

Proof There exists an expression $f=\sum_{i=1}^{m}f_{i}$ as in Definition 4.21. Let $\mathcal{I}$ denote the set of the maximal elements in the partially ordered set $\{\mathfrak{a}(f_{1},Q),\ldots,\mathfrak{a}(f_{m},Q)\}$ with $\prec_{Q}$ . There exists a decomposition

\{\mathfrak{a}(f_{1},Q),\ldots,\mathfrak{a}(f_{m},Q)\}=\coprod_{\mathfrak{a}\in\mathcal{I}}\mathcal{J}_{\mathfrak{a}}

such that any $\mathfrak{b}\in\mathcal{J}_{\mathfrak{a}}$ satisfy $\mathfrak{b}\prec_{Q}\mathfrak{a}$ . We set $f_{\mathfrak{a}}:=\sum_{\mathfrak{a}(f_{i},Q)\in\mathcal{J}_{\mathfrak{a}}}f_{i}$ . We obtain the expression $f=\sum_{\mathfrak{a}\in\mathcal{I}}f_{\mathfrak{a}}$ . It is easy to see that $f_{\mathfrak{a}}$ has a single growth order with $\mathfrak{a}(f_{\mathfrak{a}},Q)=\mathfrak{a}$ , and the first claim is proved. The second claim is clear. ∎

Note that a reduced expression $f=\sum_{\mathfrak{a}\in\mathcal{I}}f_{\mathfrak{a}}$ is not uniquely determined.

Notation 4.23.

Let $V$ be an open subset of $\widetilde{{\mathbb{C}}}$ . Let $f$ be a holomorphic function on $V\setminus\varpi^{-1}(0)$ which has multiple growth orders at any point of $V\cap\varpi^{-1}(0)$ . Then, let $\mathcal{Z}(f)$ denote the set of the points $Q\in V\cap\varpi^{-1}(0)$ such that one of the following condition is satisfied.

•

$f$ does not have single growth order at $Q$ .
•

$f$ has single growth order and turning at $Q$ .

Note that $\mathcal{Z}(f)$ is discrete in $V\cap\varpi^{-1}(0)$ .

Remark 4.24.

Let $U$ be a neighbourhood of $0$ in ${\mathbb{C}}$ . Suppose that a non-zero holomorphic function $f$ on $U\setminus\{0\}$ satisfies a linear differential equation $\partial_{z}^{n}f+\sum_{j=0}^{n-1}a_{j}(z)\partial_{z}^{j}f=0$ , where $a_{j}$ are meromorphic functions on $(U,0)$ . Then, $f$ has multiple growth orders at any point of $\varpi^{-1}(0)$ , which is a consequence of the classical asymptotic analysis. (For example, see [28, §II.1].)

4.3.5. Coordinate change

Let $U$ be an open neighbourhood of $0$ in ${\mathbb{C}}$ . Let $\gamma:U\longrightarrow{\mathbb{C}}$ be a holomorphic function such that $\gamma(0)=0$ and $\partial_{z}\gamma(0)=1$ . It induces an automorphism $\gamma^{\ast}$ of $\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ .

For any $z^{-a}$ $(a>0)$ , we have the expansion

\gamma^{\ast}(z^{-a})=z^{-a}(\gamma^{\ast}(z)/z)^{-a}=z^{-a}+\sum_{b>-a}\alpha_{a,b}z^{b}.

We define

\gamma^{\ast}_{-}(z^{-a}):=z^{-a}+\sum_{-a<b<0}\alpha_{a,b}z^{b}.

It induces an injection $\gamma_{-}^{\ast}:\mathfrak{I}(Q)\longrightarrow\mathfrak{I}(Q)$ . The following lemma is easy to see.

Lemma 4.25.

$f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has a single growth order if and only if $\gamma^{\ast}(f)$ has a single growth order. In that case, we have $\mathfrak{a}(\gamma^{\ast}(f),Q)=\gamma_{-}^{\ast}(\mathfrak{a}(f,Q))$ . As a result $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ has multiple growth orders if and only if $\gamma^{\ast}(f)$ has multiple growth orders. ∎

Let $\mathfrak{B}_{\widetilde{{\mathbb{C}}},Q}$ denote the set of $f\in\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{Q}$ with multiple growth orders. It is independent of the choice of a coordinate $z$ by Lemma 4.25.

4.3.6. Sheaf of holomorphic functions with multiple growth orders

Let $X$ be any Riemann surface with a discrete subset $D$ . Let $\varpi_{X,D}:\widetilde{X}_{D}\longrightarrow X$ denote the oriented real blowing up. Let $\iota_{X\setminus D}:X\setminus D\longrightarrow\widetilde{X}_{D}$ denote the inclusion.

Notation 4.26.

Let $\mathfrak{B}_{\widetilde{X}_{D}}\subset\iota_{X\setminus D,\ast}(\mathcal{O}_{X\setminus D})$ denote the subsheaf determined as follows.

•

$\mathfrak{B}_{\widetilde{X}_{D}|X\setminus D}=\mathcal{O}_{X\setminus D}$ .
•

For any $P\in D$ , we take a holomorphic embedding $z_{P}:X_{P}\longrightarrow{\mathbb{C}}$ of a neighbourhood $X_{P}$ of $P$ in $X$ such that $z_{P}(P)=0$ . Note that it induces the isomorphisms $\widetilde{z}_{P}:\varpi_{X,D}^{-1}(P)\simeq\varpi^{-1}(0)$ and $\widetilde{z}_{P}^{\ast}:\iota_{\ast}(\mathcal{O}_{{\mathbb{C}}^{\ast}})_{\widetilde{z}_{P}(Q)}\simeq\iota_{X\setminus D,\ast}(\mathcal{O}_{X\setminus D})_{Q}$ for any $Q\in\varpi_{X,D}^{-1}(P)$ . Then, $\mathfrak{B}_{\widetilde{X}_{D},Q}=\widetilde{z}_{P}^{-1}\bigl{(}\mathfrak{B}_{\widetilde{{\mathbb{C}}},\widetilde{z}_{P}(Q)}\bigr{)}$ holds.

If $f$ is a section of $\mathfrak{B}_{\widetilde{X}_{D}}$ on an open subset $V\subset\widetilde{X}_{D}$ , $\alpha f$ $(\alpha\in{\mathbb{C}}^{\ast})$ are also sections of $\mathfrak{B}_{\widetilde{X}_{D}}$ on $V$ . But, even if $f_{i}$ $(i=1,2)$ are sections of $\mathfrak{B}_{\widetilde{X}_{D}}$ on $V$ , $f_{1}+f_{2}$ is not necessarily a section of $\mathfrak{B}_{\widetilde{X}_{D}}$ on $V$ . For example, we set $f_{1}=e^{-z^{-1}}$ and $f_{2}=-e^{-z^{-1}}+e^{-e^{z^{-1}}}$ around $\arg(z)=0$ . Then, $f_{i}$ are sections of $\mathfrak{V}_{\widetilde{X}_{D}}$ , but $f_{1}+f_{2}=e^{-e^{z^{-1}}}$ is not.

4.3.7. Intervals with some properties

Let $f$ be a section of $\mathfrak{B}_{\widetilde{{\mathbb{C}}}}$ on an open subset $V\subset\widetilde{{\mathbb{C}}}$ .

Definition 4.27.

An open interval $I\subset\varpi^{-1}(0)\cap V$ is called positive (resp. negative) with respect to $f$ if $f$ is simply positive (resp. negative) at each point of $I\setminus\mathcal{Z}(f)$ . It is called maximal if moreover there does not exist an interval $I_{1}\subset\varpi^{-1}(0)\cap V$ such that (i) $I_{1}$ is positive (resp. negative) with respect to $f$ , (ii) $I\subsetneq I_{1}$ .

Definition 4.28.

An open interval $I\subset\varpi^{-1}(0)\cap V$ is called neutral with respect to $f$ if $f$ is neutral at each point of $I\setminus\mathcal{Z}(f)$ . It is called maximal if moreover there does not exist an interval $I_{1}\subset\varpi^{-1}(0)\cap V$ such that (i) $I_{1}$ is neutral with respect to $f$ , (ii) $I\subsetneq I_{1}$ .

Definition 4.29.

An open interval $I\subset\varpi^{-1}(0)\cap V$ is called non-positive with respect to $f$ if $f$ is neutral or simply negative at each point of $I\setminus\mathcal{Z}(f)$ . It is called maximal if moreover there does not exist an interval $I_{1}\subset\varpi^{-1}(0)\cap V$ such that (i) $I_{1}$ is non-positive with respect to $f$ , (ii) $I\subsetneq I_{1}$ .

In Definitions 4.27–4.29, if $\overline{I}$ is contained in $V$ and if $I$ is maximal, then $\partial I\subset\mathcal{Z}(f)$ .

Lemma 4.30.

If an interval $I$ is non-positive with respect to $f$ , then $I$ is either negative or neutral with respect to $f$ . Moreover, if $I$ is neutral with respect to $f$ , then $I\cap\mathcal{Z}(f)=\emptyset$ .

Proof Let $I$ be a non-positive interval. Let $I\setminus\mathcal{Z}(f)=\coprod I_{i}$ be the decomposition into the connected components. The numbering $I_{i}$ is given in the counter-clockwise way. If $f$ is negative (neutral) at a point $Q\in I_{i}$ , then $f$ is negative at any point of $I_{i}$ .

Let $Q\in I\cap\mathcal{Z}(f)$ such that $Q=\overline{I}_{i}\cap\overline{I}_{i+1}$ . There exists a reduced expression $f=\sum_{\mathfrak{a}\in\mathcal{I}}f_{\mathfrak{a}}$ at $Q$ as in Lemma 4.22. Take $Q_{i}\in I_{i}$ and $Q_{i+1}\in I_{i+1}$ . Suppose that $f$ is negative at $Q_{i}$ and neutral at $Q_{i+1}$ , and we shall deduce a contradiction. Note that $\mathfrak{a}(f,Q_{i})=\max(\mathcal{I},\prec_{Q_{i}})=:\mathfrak{b}$ and $\mathfrak{a}(f,Q_{i+1})=\max(\mathcal{I},\prec_{Q_{i+1}})=0$ . We obtain $|\mathfrak{b}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{b})(Q)=0$ . If $Q_{i+1}$ is sufficiently close to $Q$ , we obtain $|\mathfrak{b}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{b})(Q_{i+1})>0$ , i.e., $0\prec_{Q_{i+1}}\mathfrak{b}$ . It contradicts $\mathfrak{b}\prec_{Q_{i+1}}0$ . Therefore, if $f$ is negative at $Q_{i}$ , then $f$ is negative at $Q_{i+1}$ . If $f$ is neutral at $Q_{i}$ , then $f$ is neutral at $Q_{i+1}$ , and moreover we obtain $\mathcal{I}=\{0\}$ . Then, we easily obtain the claim of the lemma. ∎

Lemma 4.31.

Suppose that $I$ is negative with respect to $f$ such that $I\cap\mathcal{Z}(f)$ consists of one point $Q$ . Choose $Q_{1}$ and $Q_{2}$ from the two connected components of $I\setminus\{Q\}$ . Let $f=\sum_{\mathfrak{a}\in\mathcal{I}}f_{\mathfrak{a}}$ be a reduced expression at $Q$ . Then, the following holds.

•

$\deg\mathfrak{a}(f,Q_{1})=\deg\mathfrak{a}(f,Q_{2})$ .
•

For any $\mathfrak{a}\in\mathcal{I}$ , we obtain $\deg(\mathfrak{a})=\deg(\mathfrak{a}(f,Q_{i}))$ . Moreover, $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})(Q)<0$ .

Proof Note that $\mathfrak{a}(f,Q_{i})\in\mathcal{I}$ . Let us prove

\deg\mathfrak{a}(f,Q_{1})\geq\deg\mathfrak{a}(f,Q_{2}).

(50)

Suppose $\deg\mathfrak{a}(f,Q_{1})<\deg\mathfrak{a}(f,Q_{2})$ , and we shall deduce a contradiction. If $|\mathfrak{a}(f,Q_{2})|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q_{2}))(Q)<0$ , we obtain $\mathfrak{a}(f,Q_{2})\prec_{Q}\mathfrak{a}(f,Q_{1})$ , and hence $\mathfrak{a}(f,Q_{2})\prec_{Q_{2}}\mathfrak{a}(f,Q_{1})$ holds, which contradicts $\mathfrak{a}(f,Q_{2})=\max(\mathcal{I},\prec_{Q_{2}})$ . If $|\mathfrak{a}(f,Q_{2})|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q_{2}))(Q)\geq 0$ , because $\mathfrak{a}(f,Q_{1})\prec_{Q_{2}}\mathfrak{a}(f,Q_{2})$ , we obtain $|\mathfrak{a}(f,Q_{2})|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q_{2}))(Q_{2})>0$ , which contradicts that $f$ is negative at $Q_{2}$ . Thus, we obtain (50). Similarly, we obtain $\deg\mathfrak{a}(f,Q_{1})\leq\deg\mathfrak{a}(f,Q_{2})$ , and hence $\deg\mathfrak{a}(f,Q_{1})=\deg\mathfrak{a}(f,Q_{2})$ .

Take $\mathfrak{a}\in\mathcal{I}$ . Let us prove $\deg(\mathfrak{a})=\deg(\mathfrak{a}(f,Q_{i}))$ . Suppose $\deg\mathfrak{a}>\deg\mathfrak{a}(f,Q_{i})$ . If $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})(Q)<0$ , we obtain $\mathfrak{a}\prec_{Q}\mathfrak{a}(f,Q_{i})$ , which contradicts that the expression $f=\sum_{\mathfrak{a}\in\mathcal{I}}f_{\mathfrak{a}}$ is reduced. If $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})(Q)\geq 0$ , then either $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})(Q_{1})>0$ or $|\mathfrak{a}|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a})(Q_{2})>0$ holds, which contradicts that $f$ is negative at $Q_{i}$ . Suppose $\deg\mathfrak{a}<\deg\mathfrak{a}(f,Q_{i})$ . Because $|\mathfrak{a}(f,Q_{i})|^{-1}\mathop{\rm Re}\nolimits(\mathfrak{a}(f,Q_{i}))(Q_{i})<0$ , we obtain $\mathfrak{a}(f,Q_{i})\prec_{Q_{i}}\mathfrak{a}$ , which contradicts $\mathfrak{a}(f,Q_{i})=\max(\mathcal{I},\prec_{Q_{i}})$ . In all, we obtain $\deg(\mathfrak{a})=\deg(\mathfrak{a}(f,Q_{i}))$ . ∎

Recall that we identify $\widetilde{{\mathbb{C}}}\simeq{\mathbb{R}}_{\geq 0}\times S^{1}$ by the polar coordinate system. For any $a<b$ , we set $]a,b[=\{\theta\in{\mathbb{R}}\,|\,a<\theta<b\}$ . If $b-a<2\pi$ , we may naturally regard it as an interval in $S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$ .

Definition 4.32.

An open interval $I\subset\varpi^{-1}(0)\cap V$ is called special with respect to $f$ if the following holds.

•

The length of $I$ is $\pi/\rho$ for $\rho>1/2$ . Namely, $I=\{0\}\times]\theta_{1},\theta_{1}+\pi/\rho[$ for some $\theta_{1}$ .

•

There exists $\alpha\in{\mathbb{C}}\setminus\{0\}$ such that (i) $\deg(\mathfrak{a}(f,Q)-\alpha z^{-\rho})<\deg(\mathfrak{a}(f,Q))=\rho$ for any $Q\in I\setminus\mathcal{Z}(f)$ , (ii) $\mathop{\rm Re}\nolimits(\alpha e^{-\rho\sqrt{-1}\theta})<0$ on $I$ , which particularly implies that

\mathop{\rm Re}\nolimits(\alpha e^{-\rho\sqrt{-1}\theta_{1}})=\mathop{\rm Re}\nolimits(\alpha e^{-\rho\sqrt{-1}(\theta_{1}+\pi/\rho)})=0.

Lemma 4.33.

Let $I$ be an interval which is negative with respect to $f$ . Note that $\rho:=\deg(\mathfrak{a}(f,Q))$ $(Q\in I\setminus\mathcal{Z}(f))$ is well defined. Then, either one of the following holds.

•

$I$ is special with respect to $f$ .
•

The length of $I$ is strictly smaller than $\pi/\rho$ .

Proof There exists an interval $\widetilde{I}=\{\psi_{0}\leq\theta\leq\psi_{1}\}\subset{\mathbb{R}}$ such that the map $\theta\longmapsto e^{\sqrt{-1}\theta}$ induces a diffeomorphism $\widetilde{I}\simeq I$ . Let $\psi_{0}<\theta_{1}<\cdots<\theta_{\ell}<\psi_{1}$ be determined by $\{(0,e^{\sqrt{-1}\theta_{i}})\}=I\cap\mathcal{Z}(f)$ . We set $\theta_{0}:=\psi_{0}$ and $\theta_{\ell+1}:=\psi_{1}$ . We set $\widetilde{I}_{i}=\{0\}\times]\theta_{i-1},\theta_{i}[\subset\widetilde{I}$ $(i=1,\ldots,\ell+1)$ . We obtain the corresponding subsets $I_{i}\subset I$ . Choose $Q_{i}\in I_{i}$ . There exist $\alpha_{i}\in{\mathbb{C}}^{\ast}$ such that $\deg(\mathfrak{a}(f,Q_{i})-\alpha_{i}z^{-\rho})<\deg(\mathfrak{a}(f,Q_{i}))$ . There exist intervals $J_{i}=]\varphi_{i},\varphi_{i}+\pi/\rho[$ such that (i) $\varphi_{i}\leq\theta_{i-1}<\theta_{i}\leq\varphi_{i}+\pi/\rho$ , (ii) $\mathop{\rm Re}\nolimits(\alpha_{i}e^{-\rho\sqrt{-1}\theta})<0$ on $J_{i}$ .

For $1\leq i\leq\ell$ , we obtain $\mathop{\rm Re}\nolimits(\alpha_{i}e^{-\rho\sqrt{-1}\theta_{i}})=\mathop{\rm Re}\nolimits(\alpha_{i+1}e^{-\rho\sqrt{-1}\theta_{i}})<0$ . We also obtain

\mathop{\rm Re}\nolimits(\alpha_{i}e^{-\rho\sqrt{-1}(\theta_{i}+\epsilon)})<\mathop{\rm Re}\nolimits(\alpha_{i+1}e^{-\rho\sqrt{-1}(\theta_{i}+\epsilon)}),

\mathop{\rm Re}\nolimits(\alpha_{i}e^{-\rho\sqrt{-1}(\theta_{i}-\epsilon)})>\mathop{\rm Re}\nolimits(\alpha_{i+1}e^{-\rho\sqrt{-1}(\theta_{i}-\epsilon)})

for any sufficiently small positive number $\epsilon$ . Therefore, we obtain $\varphi_{i+1}\leq\varphi_{i}$ . Moreover, if $\varphi_{i+1}=\varphi_{i}$ , there exists $a>0$ such that $\alpha_{i+1}=a\alpha_{i}$ . Because $\mathop{\rm Re}\nolimits(\alpha_{i}e^{-\rho\sqrt{-1}\theta_{i}})=\mathop{\rm Re}\nolimits(\alpha_{i+1}e^{-\rho\sqrt{-1}\theta_{i}})<0$ , we obtain $\alpha_{i+1}=\alpha_{i}$ . Therefore, we obtain

\theta_{\ell+1}\leq\varphi_{\ell+1}+\pi/\rho\leq\cdots\leq\varphi_{1}+\pi/\rho\leq\theta_{0}+\pi/\rho.

Namely, we obtain $\theta_{\ell+1}-\theta_{0}\leq\pi/\rho$ . If $\theta_{\ell+1}=\theta_{0}+\pi/\rho$ , we obtain $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{\ell+1}$ . ∎

We also obtain the following lemma.

Lemma 4.34.

Suppose that $V$ is a neighbourhood of $\varpi^{-1}(0)$ , and that $f$ is non-positive at each point of $\varpi^{-1}(0)\setminus\mathcal{Z}(f)$ . Then, $f$ is meromorphic at $0$ .

Proof If $f$ is neutral at a point of $\varpi^{-1}(0)\setminus\mathcal{Z}(f)$ , we obtain that $f$ is neutral at any point of $\varpi^{-1}(0)\setminus\mathcal{Z}(f)$ according to Lemma 4.30, and hence we obtain that $f$ is meromorphic. Suppose that $f$ is simply negative at any point of $\varpi^{-1}(0)\setminus\mathcal{Z}(f)$ . By Lemma 4.31, there exist $\epsilon>0$ and $\rho>0$ such that $|f|=O\bigl{(}e^{-\epsilon|z|^{-\rho}}\bigr{)}$ on $V\setminus\varpi^{-1}(0)$ , which implies $f=0$ . ∎

4.3.8. Estimates in a single case

For any $R>0$ and $0<L<\pi$ , we set $W(R,L):=\bigl{\{}w\in{\mathbb{C}}\,\big{|}\,|w|>R,\,|\arg(w)|<L\bigr{\}}$ . Let $\widetilde{\mathbb{P}}^{1}_{\infty}\longrightarrow\mathbb{P}^{1}$ be the oriented real blowing up at $\infty$ . We regard $W(R,L)$ as an open subset of $\widetilde{\mathbb{P}}^{1}_{\infty}$ . Let $\overline{W}(R,L)$ denote the closure of $W(R,L)$ in $\widetilde{\mathbb{P}}^{1}_{\infty}$ . Let $Q_{0}\in\overline{W}(R,L)$ denote the point corresponding to $+\infty$ , i.e., $Q_{0}$ is the limit of any sequence of positive numbers $t_{i}$ in $\widetilde{\mathbb{P}}^{1}_{\infty}$ with $t_{i}\to\infty$ . Let $\Phi_{\beta,b}:W(R,L)\longrightarrow{\mathbb{C}}^{\ast}$ be given by $\Phi_{\beta,b}(w)=\beta w^{-b}$ for a positive number $b$ and a non-zero complex number $\beta$ . It induces $\widetilde{\Phi}_{\beta,b}:\overline{W}(R,L)\longrightarrow\widetilde{{\mathbb{C}}}$ .

Let $f$ be a section of $\mathfrak{B}_{\widetilde{{\mathbb{C}}}}$ on an open subset $V\subset\widetilde{{\mathbb{C}}}$ . Suppose that $f$ has a single growth order at $Q\in V\cap\varpi^{-1}(0)$ . Let $0<L<\widetilde{L}<\pi$ . By choosing $\beta\in{\mathbb{C}}^{\ast}$ , $b>0$ and $R>0$ appropriately, we obtain $\widetilde{\Phi}_{\beta,b}:W(R,\widetilde{L})\to\widetilde{{\mathbb{C}}}$ such that $\widetilde{\Phi}_{\beta,b}(Q_{0})=Q$ and $\widetilde{\Phi}_{\beta,b}(\overline{W}(R,\widetilde{L}))\subset V$ . We obtain the holomorphic function $\Phi_{b,\beta}^{\ast}(f)$ on $W(R,\widetilde{L})$ which induces a section of $\mathfrak{B}_{\widetilde{\mathbb{P}}^{1}_{\infty}}$ on $\overline{W}(R,\widetilde{L})$ . If $b$ is sufficiently small and $R$ is sufficiently large,

e^{-\mathfrak{a}(f,Q)}z^{-a(f,Q)}(\log z)^{-j(f,Q)}f

is regularly bounded on $\widetilde{\Phi}_{\beta,b}(\overline{W}(R,\widetilde{L}))\setminus\varpi^{-1}(0)$ .

Lemma 4.35.

For any $0<L_{2}<L_{1}<L$ , there exist $R_{2}>R_{1}>R$ and subsets $\mathcal{C}_{1}\subset\mathcal{C}_{2}\subset W(R,L)$ such that the following holds.

•

There exists $C>0$ such that

\bigl{|}\Phi_{\beta,b}^{\ast}(f)\bigr{|}\geq Ce^{\mathop{\rm Re}\nolimits\Phi_{\beta,b}^{\ast}\mathfrak{a}(f,Q)}|w|^{ba(f,Q)}(\log|w|)^{j(f,Q)}

on $W(R,L)\setminus\mathcal{C}_{1}$ .

•

Let $\mathcal{D}$ be any connected component of $\mathcal{C}_{2}$ such that

$\mathcal{D}\cap W(R_{2},L_{2})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $W(R_{1},L_{1})$ .
•

For any $w_{0}\in W(R_{1},L_{1})\setminus\mathcal{C}_{2}$ , we obtain $\{|w-w_{0}|<1\}\subset W(R,L)\setminus\mathcal{C}_{1}$ .

Proof It is enough to prove the case where $\mathfrak{a}(f,Q)=0$ , $a(f,Q)=0$ and $j(f,Q)=0$ , i.e., $f$ is regularly bounded at $Q$ .

For any $B_{1}\in{\mathbb{R}}$ and $B_{2}>0$ , we set $\widetilde{W}(B_{1},B_{2}):=\{\zeta\in{\mathbb{C}}\,|\,\mathop{\rm Re}\nolimits(\zeta)>B_{1},\,|\mathop{\rm Im}\nolimits(\zeta)|<B_{2}\}$ . Let $\Phi_{1}:{\mathbb{C}}\longrightarrow{\mathbb{C}}$ be given by $\Phi_{1}(\zeta)=e^{\zeta}$ . It induces $\widetilde{W}(B_{1},B_{2})\simeq W(e^{B_{1}},B_{2})$ for any $B_{1},B_{2}>0$ . We set $\widetilde{f}:=\Phi_{1}^{\ast}\Phi_{\beta,b}^{\ast}(f)$ on $W(\log R,\widetilde{L})$ . Let $Z(\widetilde{f})$ denote the zero set of $\widetilde{f}$ . By Proposition 4.9, there exists $N>0$ such that the following holds for any $x_{1}>\log R$ :

\#\Bigl{(}Z(\widetilde{f})\cap\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,x_{1}<\mathop{\rm Re}\nolimits(\zeta)<x_{1}+L,\,\,|\mathop{\rm Im}\nolimits(\zeta)|<L\bigr{\}}\Bigr{)}<N.

Take

0<\delta<\frac{1}{10^{3}N}\min\bigl{\{}|L-L_{1}|,|L_{1}-L_{2}|,|L_{2}|\bigr{\}}.

We set

\mathcal{B}_{1}:=\bigcup_{\zeta_{1}\in Z(\widetilde{f})}\{|\zeta-\zeta_{1}|<\delta\}\cap\widetilde{W}(\log R,L),

\mathcal{B}_{2}:=\bigcup_{\zeta_{1}\in Z(\widetilde{f})}\{|\zeta-\zeta_{1}|<3\delta\}\cap\widetilde{W}(\log R,L).

There exists $C>0$ such that $|\widetilde{f}|>C$ on $\widetilde{W}(\log R,L)\setminus\mathcal{B}_{1}$ by Corollary 4.11.

We set $\mathcal{C}_{i}=\Phi_{1}(\mathcal{B}_{i})\subset W(R,L)$ . Then, the first condition is satisfied. Take $R_{1}>\max\{Re^{L},10\delta^{-1}\}$ . For any $\zeta_{1},\zeta_{2}\in\widetilde{W}(\log R)$ , we have

|\zeta_{1}-\zeta_{2}|\leq|e^{\zeta_{1}}-e^{\zeta_{2}}|\max\{e^{-\mathop{\rm Re}\nolimits(\zeta_{1})},e^{-\mathop{\rm Re}\nolimits(\zeta_{2})}\}.

Then, the third condition is satisfied. Take $R_{2}>R_{1}e^{L}$ . Let $\widetilde{\mathcal{D}}$ be a connected component of $\mathcal{B}_{2}$ such that $\widetilde{\mathcal{D}}\cap\widetilde{W}(\log R_{2},L_{2})\neq\emptyset$ . Then, there exists a subset $Z_{\widetilde{\mathcal{D}}}(\widetilde{f})\subset Z(\widetilde{f})$ such that $\widetilde{\mathcal{D}}$ is the union of $\{|\zeta-\zeta_{j}|<3\delta\}$ $(\zeta_{j}\in Z_{\widetilde{\mathcal{D}}}(\widetilde{f}))$ . Let us observe that $\#Z_{\widetilde{\mathcal{D}}}(\widetilde{f})\leq N$ . Take $\zeta_{0}\in Z_{\widetilde{\mathcal{D}}}(\widetilde{f})$ . If $\#Z_{\widetilde{\mathcal{D}}}(\widetilde{f})>N$ , there exists $\eta\in Z_{\widetilde{\mathcal{D}}}(\widetilde{f})$ such that $|\mathop{\rm Re}\nolimits(\eta-\zeta_{0})|>L/2$ . There exists a sequence $\zeta_{1},\ldots,\zeta_{m},\zeta_{m+1}=\eta\in Z_{\widetilde{\mathcal{D}}}(\widetilde{f})$ such that $d(\zeta_{i},\zeta_{i+1})<3\delta$ for $i=0,\ldots,m$ . There exists $i_{0}$ such that $|\mathop{\rm Re}\nolimits(\zeta_{0}-\zeta_{i})|\leq L/2$ for $i<i_{0}$ and $|\mathop{\rm Re}\nolimits(\zeta_{0}-\zeta_{i_{0}})|>L/2$ . Note that $i_{0}\leq N$ . Because of the choice of $\delta$ , we obtain $d(\zeta_{0},\zeta_{i_{0}})<L/2$ . Thus, we arrive at a contradiction, and hence we obtain $\#Z_{\widetilde{\mathcal{D}}}(\widetilde{f})\leq N$ . Then, we obtain that $\widetilde{\mathcal{D}}$ is relatively compact in $\widetilde{W}(\log R_{1},L_{1})$ , and hence the second condition is satisfied. ∎

We set $g=w+\alpha w^{c}$ on $W(R,\widetilde{L})$ for $\alpha\in{\mathbb{C}}\setminus\{0\}$ and $0<c<1$ . For any $B^{(1)}\in{\mathbb{R}}$ and $0<L^{(1)}<L$ , there exists $R^{(1)}\geq R$ such that

S(g,R^{(1)},B^{(1)},L^{(1)})=\\ \bigl{\{}w\in W(R,\widetilde{L})\,\big{|}\,|g(w)|>R^{(1)},\,\,\mathop{\rm Im}\nolimits(g(w))\geq B^{(1)},\,\,\arg(g(w))<L^{(1)}\bigr{\}}\\ \subset W(R,L).

(51)

Let $Z(\Phi_{b,\beta}^{\ast}(f))$ denote the zero set of $\Phi_{b,\beta}^{\ast}(f)$ .

Lemma 4.36.

There exists $N>0$ such that the following holds for any $M>R^{(1)}e^{L}$ :

\#\Bigl{(}Z(\Phi_{b,\beta}^{\ast}(f))\cap\bigl{\{}w\in S(g,R^{(1)},B^{(1)},L^{(1)})\,\big{|}\,M<|g(w)|<Me^{L}\bigr{\}}\Bigr{)}<N.

Proof There exist $M_{0}>R$ and $\epsilon>0$ such that the following holds for any $M>M_{0}$ :

\bigl{\{}w\in S(g,R^{(1)},B^{(1)},L^{(1)})\,\big{|}\,M<|g(w)|<Me^{L}\bigr{\}}\subset\\ \bigl{\{}w\in W(R,L)\,\big{|}\,Me^{-\epsilon}<|w|<Me^{L+\epsilon}\bigr{\}}.

(52)

Then, the claim follows from Proposition 4.9. ∎

Lemma 4.37.

For any $\rho>0$ and $\kappa>0$ , there exist $R_{0}^{(1)}>0$ , $C_{1}>0$ and $\kappa_{1}>0$ such that

e^{-\mathop{\rm Re}\nolimits\Phi_{\beta,b}^{\ast}\mathfrak{a}(f,Q)}|w|^{-ba(f,Q)}(\log|w|)^{-j(f,Q)}\bigl{|}\Phi_{\beta,b}^{\ast}(f)(w)\bigr{|}\geq C_{1}e^{-\kappa_{1}|w|^{\rho}}

for any $w\in S(g,R_{0}^{(1)},B^{(1)},L^{(1)})$ such that $d(w,Z(\Phi_{\beta,b}^{\ast}(f)))\geq\frac{1}{2}e^{-\kappa|w|^{\rho}}$ .

Proof We have only to study the case where $\mathfrak{a}(f,Q)=0$ , $a(f,Q)=0$ and $j(f,Q)=0$ . We use the notation in the proof of Lemma 4.35.

For $\zeta_{1},\zeta_{2}\in{\mathbb{C}}$ with $|\zeta_{1}-\zeta_{2}|\leq 1$ , we have $|e^{\zeta_{1}}-e^{\zeta_{2}}|\leq e^{\mathop{\rm Re}\nolimits(\zeta_{1})+1}|\zeta_{1}-\zeta_{2}|$ . Hence, there exists $C_{2}>0$ such that the following holds for any $w\in S(g,R^{(1)},B^{(1)},L^{(1)})$ satisfying $d(w,Z(\Phi_{\beta,b}^{\ast}(f)))\geq\frac{1}{2}e^{-\kappa|w|^{\rho}}$ .

•

Take $\zeta\in\widetilde{W}(\log R,L)$ such that $e^{\zeta}=w$ . Then, we obtain $d(\zeta,Z(\widetilde{f}))>C_{2}|w|^{-1}e^{-\kappa|w|^{\rho}}$ .

By Corollary 4.11, there exist constants $C_{i}>0$ $(i=3,4,5,6)$ such that the following holds for any $w\in S(g,R^{(1)},B^{(1)},L^{(1)})$ satisfying $d(w,Z(\Phi_{\beta,b}^{\ast}(f)))\geq\frac{1}{2}e^{-\kappa|w|^{\rho}}$ :

\bigl{|}\Phi_{\beta,b}^{\ast}(f)(w)\bigr{|}\geq C_{3}(|w|^{-1}e^{-\kappa|w|^{\rho}})^{C_{4}}\geq C_{5}e^{-C_{6}\kappa|w|^{\rho}}.

Thus, we obtain the claim of the lemma. ∎

Lemma 4.38.

For any $B^{(1)}_{1}>B^{(1)}$ , $0<L^{(1)}_{1}<L^{(1)}$ and $\rho>0$ , there exist $R^{(1)}_{1}>R^{(1)}$ , and a subset $\mathcal{C}\subset S(g,R^{(1)},B^{(1)},L^{(1)})$ such that the following holds:

•

There exist $C_{i}>0$ $(i=1,2)$ such that

e^{-\mathop{\rm Re}\nolimits(\Phi_{\beta,b}^{\ast}\mathfrak{a}(f,Q))}|w|^{-ba(f,Q)}(\log|w|)^{-j(f,Q)}\bigl{|}\Phi_{\beta,b}^{\ast}(f)(w)\bigr{|}\geq\\ C_{1}\exp(-C_{2}|w|^{\rho})

(53)

on $S(g,R^{(1)},B^{(1)},L^{(1)})\setminus\mathcal{C}$ .

•

Let $\mathcal{D}$ be a connected component of $\mathcal{C}$ such that

$\mathcal{D}\cap S(g,R_{1}^{(1)},B^{(1)}_{1},L^{(1)}_{1})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $S(g,R^{(1)},B^{(1)},L^{(1)})$ .
•

There exist an increasing sequence of positive numbers $T_{i}$ such that (i) $\lim T_{i}=\infty$ , (ii) $\mathcal{C}\cap\{|g|=T_{i}\}\cap S(g,R^{(1)},B^{(1)},L^{(1)})=\emptyset$ .

Proof We have only to study the case where $\mathfrak{a}(f,Q)=0$ , $a(f,Q)=0$ and $j(f,Q)=0$ . Let $N$ be as in Lemma 4.36. Take $\kappa>0$ satisfying

10^{3}Ne^{-\kappa(R^{(1)})^{\rho}}<\\ \min\Bigl{\{}|B^{(1)}_{1}-B^{(1)}|,e^{R^{(1)}}\!\!\!\sin\bigl{(}(L^{(1)}-L_{1}^{(1)})/2\bigr{)},e^{R^{(1)}}\!\!\!\sin(L_{1}^{(1)}/2),R^{(1)}(e^{L}-1)\Bigr{\}}.

(54)

We set

\mathcal{C}:=\bigcup_{w_{1}\in Z(\Phi_{\beta,b}^{\ast}(f))}\bigl{\{}|w-w_{1}|<e^{-\kappa|w_{1}|^{\rho}}\bigr{\}}\cap S(g,R^{(1)},B^{(1)},L^{(1)}).

For $0<\nu<1$ and for $w,a\in{\mathbb{C}}$ with $|w|>1$ and $|a|<\nu$ , we have

\bigl{|}|w+a|^{\rho}-|w|^{\rho}\bigr{|}\leq\rho\nu\left(|w|+\nu\right)^{\rho-1}+\rho\nu\left(|w|-\nu\right)^{\rho-1}.

Let $R_{0}^{(1)}\geq R^{(1)}$ be as in Lemma 4.37. There exists $R_{2}^{(1)}>R_{0}^{(1)}e^{L}+1$ such that the following holds for any $w\in{\mathbb{C}}$ with $|w|>R_{1}^{(1)}$ :

\frac{1}{2}\exp\Bigl{(}\kappa\rho e^{-\kappa|w|^{\rho}}\bigl{(}(|w|+1)^{\rho-1}+(|w|-1)^{\rho-1}+2|w|^{\rho-1}\bigr{)}\Bigr{)}\leq 3/4.

For $w_{2},w_{3}\in{\mathbb{C}}$ satisfying $|w_{i}|>R_{2}^{(1)}$ satisfying $|w_{2}-w_{3}|\leq\frac{1}{2}e^{-\kappa|w_{2}|^{\rho}}$ , we obtain $|w_{2}-w_{3}|\leq(3/4)e^{-\kappa|w_{3}|^{\rho}}$ . Hence, by Lemma 4.37, there exist $C^{\prime}_{i}>0$ $(i=1,2)$ such that

e^{-\mathop{\rm Re}\nolimits(\Phi_{\beta,b}^{\ast}\mathfrak{a}(f,Q))}|w|^{-ba(f,Q)}(\log|w|)^{-j(f,Q)}\bigl{|}\Phi_{\beta,b}^{\ast}(f)(w)\bigr{|}\geq\\ C^{\prime}_{1}\exp(-C_{2}^{\prime}|w|^{\rho})

(55)

on $S(g,R_{2}^{(1)},B^{(1)},L^{(1)})\setminus\mathcal{C}$ . Because

S(g,R^{(1)},B^{(1)},L^{(1)})\setminus S(g,R^{(1)}_{1},B^{(1)},L^{(1)})

is relatively compact, we obtain the first claim. Take $R^{(1)}_{1}>R^{(1)}e^{L}$ . We can check the second claim by using the argument in the proof of Lemma 4.35. By Lemma 4.36, there exists an infinite sequence $T_{i}$ as desired. ∎

4.3.9. Estimates in a multiple case

Let $f$ be a section of $\mathfrak{B}_{\widetilde{{\mathbb{C}}}}$ on an open subset $V\subset\widetilde{{\mathbb{C}}}$ . Let $Q\in V\cap\varpi^{-1}(0)$ .

Lemma 4.39.

There exist $\mathfrak{a}\in\mathfrak{I}(Q)=\bigoplus_{b>0}{\mathbb{C}}z^{-b}$ , positive constants $C_{i}$ $(i=1,2)$ and a neighbourhood $\mathcal{U}$ of $Q$ in $V$ such that

|f|\leq C_{1}|z|^{-C_{2}}e^{\mathop{\rm Re}\nolimits\mathfrak{a}}

on $\mathcal{U}\setminus\varpi^{-1}(0)$ . If $f$ has an expression $f=\sum f_{j}$ such that each $f_{j}$ is simply negative, then we may assume $\mathfrak{a}\prec_{Q}0$ .

Proof Let $f=\sum f_{j}$ be an expression such that each $f_{j}$ has a single growth order at $Q$ . There exists $\mathfrak{a}\in\mathfrak{I}(Q)$ such that $\mathfrak{a}(f_{j},Q)\prec_{Q}\mathfrak{a}$ for any $j$ . There exists $C_{2}>0$ such that $C_{2}>\max\{\mathfrak{a}(f_{j},Q)\}$ for any $j$ . (See Definition 4.17 for $\mathfrak{a}(f_{j},Q)$ .) Then, there exist $C_{3}>0$ and a neighbourhood $\mathcal{U}$ of $Q$ in $V$ such that $|f_{j}|\leq C_{3}|z|^{-C_{2}}e^{\mathop{\rm Re}\nolimits(\mathfrak{a})}$ on $\mathcal{U}\setminus\varpi^{-1}(0)$ for any $j$ . Then, we obtain the claim of the lemma. ∎

Let us state a lower bound of $f$ around $Q$ outside an exceptional subset. We identify $\widetilde{{\mathbb{C}}}={\mathbb{R}}_{\geq 0}\times S^{1}$ by the polar decomposition $z=re^{\sqrt{-1}\theta}$ . We choose $\theta_{0}$ such that $Q$ is expressed as $(0,e^{\sqrt{-1}\theta_{0}})$ . Suppose that $f$ is expressed as a sum $\sum_{i=1}^{m}f_{i}$ on

V_{0}:=\Bigl{\{}(r,e^{\sqrt{-1}\theta})\,\Big{|}\,|\theta-\theta_{0}|<\frac{\pi}{2}\kappa_{0},\,\,\,0<r<r_{0}\Bigr{\}}\subset V,

where $f_{i}$ are holomorphic functions with a single growth order, and $\kappa_{0}$ and $r_{0}$ are positive constants.

Lemma 4.40.

Take $0<\kappa<\kappa_{0}$ such that $\max\{\kappa\deg\mathfrak{a}(f_{i},Q)\}<1$ . There exist a neighbourhood $\mathcal{U}$ of $Q$ in $\widetilde{{\mathbb{C}}}$ and a subset $Z\subset\mathcal{U}\setminus\varpi^{-1}(0)$ such that the following holds.

•

There exists a subset $Z_{1}\subset{\mathbb{R}}_{>0}$ with $\int_{Z_{1}}dt/t<\infty$ such that $Z\subset Z_{1}\times S^{1}$ .
•

For any $\delta>0$ , there exists $C>0$ such that the following holds on $\mathcal{U}\setminus(\varpi^{-1}(0)\cup Z)$ :

$|f|\geq C\exp\Bigl{(}-\delta|z|^{-1/\kappa}\Bigr{)}.$

Proof We may assume that $\theta_{0}=0$ . We set

W_{2}:=\{w\in{\mathbb{C}}\,|\,\mathop{\rm Re}\nolimits(w)\geq 0\}.

We define the map $\Psi:W_{2}\longrightarrow{\mathbb{C}}$ by $\Psi(w)=(w+C_{1})^{-\kappa}$ for some sufficiently large $C_{1}>0$ such that the image of $\Psi$ is contained in $V_{0}$ . We obtain the expression

\Psi^{\ast}(f)=\sum_{i=1}^{m}\Psi^{\ast}(f_{i})

on $W_{2}$ . There exist $C_{2}>0$ such that

\bigl{|}\Psi^{\ast}(f_{i})\bigr{|}\leq C_{2}e^{\mathop{\rm Re}\nolimits(\Psi^{\ast}\mathfrak{a}(f_{i},Q))}|w+C_{1}|^{\kappa\cdot a(f_{i},Q)}(\log|w+C_{1}|)^{j(f_{i},Q)}

on $W_{2}$ . Take $b$ such that $\max\{\kappa\deg(\mathfrak{a}(f_{i},Q))\}<b<1$ . There exists $A>0$ such that the following holds on $W_{2}$ :

\mathop{\rm Re}\nolimits(\Psi^{\ast}\mathfrak{a}(f_{i},Q))+\log\bigl{(}|w+C_{1}|^{\kappa\cdot a(f_{i},Q)}(\log|w+C_{1}|)^{j(f_{i},Q)}\bigr{)}\\ \leq A\mathop{\rm Re}\nolimits\bigl{(}(w+C_{1})^{b}\bigr{)}.

(56)

Hence, there exists $B>0$ such that

\log|\Psi^{\ast}(f)|\leq A\mathop{\rm Re}\nolimits\Bigl{(}(w+C_{1})^{b}\Bigr{)}+B.

(57)

Let $a_{1},a_{2},\ldots$ be the zeroes of $\Psi^{\ast}(f)$ in $W_{2}$ . According to [22, §14.2, Theorem 2], we obtain

\sum_{j=1}^{\infty}\frac{\mathop{\rm Re}\nolimits(a_{j})}{1+|a_{j}|^{2}}<\infty.

According to [22, §14.2, Theorem 3], we obtain the following description of $\log|\Psi^{\ast}(f)|$ :

\log|\Psi^{\ast}(f)(w)|=\sum_{j=1}^{\infty}\log\left|\frac{w-a_{j}}{w+\overline{a}_{j}}\right|\\ +\frac{\mathop{\rm Re}\nolimits(w)}{\pi}\int_{-\infty}^{\infty}\frac{d\nu_{1}(t)}{|t-\sqrt{-1}w|^{2}}-\frac{\mathop{\rm Re}\nolimits(w)}{\pi}\int_{-\infty}^{\infty}\frac{d\nu_{2}(t)}{|t-\sqrt{-1}w|^{2}}+\sigma\mathop{\rm Re}\nolimits(w).

(58)

Here, $\sigma$ is a real number, and $\nu_{i}$ are non-negative measures on ${\mathbb{R}}$ such that

\int_{-\infty}^{\infty}\frac{d\nu_{i}(t)}{1+t^{2}}<\infty.

According to the Hayman theorem [22, §15, Theorem 1], there exist a sequence $w_{1},w_{2},\ldots$ in $\{w\in{\mathbb{C}}\,|\,\mathop{\rm Re}\nolimits(w)>0\}$ and a sequence of positive numbers $\rho_{1},\rho_{2},\ldots$ such that $\sum\rho_{j}/|w_{j}|<\infty$ and that

\left|\sum_{j=1}^{\infty}\log\left|\frac{w-a_{j}}{w+\overline{a}_{j}}\right|\right|+\left|\frac{\mathop{\rm Re}\nolimits(w)}{\pi}\int_{-\infty}^{\infty}\frac{d\nu_{1}(t)}{|t-\sqrt{-1}w|^{2}}\right|\\ +\left|\frac{\mathop{\rm Re}\nolimits(w)}{\pi}\int_{-\infty}^{\infty}\frac{d\nu_{2}(t)}{|t-\sqrt{-1}w|^{2}}\right|=o(|w|)

(59)

outside of $\bigcup\bigl{\{}w\,\big{|}\,|w-w_{j}|<\rho_{j}\bigr{\}}$ . By (57), we obtain that $\sigma=0$ , and hence $\log\bigl{|}\Psi^{\ast}(f)\bigr{|}=o\bigl{(}|w|\bigr{)}$ outside of $\bigcup_{j=1}^{\infty}\bigl{\{}w\,\big{|}\,|w-w_{j}|<\rho_{j}\bigr{\}}$ . We note that $\{|w-w_{j}|<\rho_{j}\}\subset\{|w_{j}|-\rho_{j}<|w|<|w_{j}|+\rho_{j}\}$ . Because $\sum\rho_{j}/|w_{j}|<\infty$ , we have $\rho_{j}/|w_{j}|\to 0$ as $j\to\infty$ , and there exists $A>0$ such that

\int_{|w_{j}|-\rho_{j}}^{|w_{j}|+\rho_{j}}dt/t=\log\left(\frac{|w_{j}|+\rho_{j}}{|w_{j}|-\rho_{j}}\right)\leq A\frac{\rho_{j}}{|w_{j}|}.

Then, the claim of the lemma follows. ∎

5. Cyclic Higgs bundles with multiple growth orders on sectors

5.1. Statements

Let $U$ be a neighbourhood of $0$ in ${\mathbb{C}}$ . Let $\varpi:\widetilde{U}\longrightarrow U$ be the oriented real blowing up. Let $V$ be an open subset of $\widetilde{U}$ . Let $f$ be a section of $\mathfrak{B}_{\widetilde{U}}$ on $V$ . We set $q=f\cdot(dz)^{r}$ .

5.1.1. Positive intervals

Let $I\subset\varpi^{-1}(0)\cap V$ be a positive interval with respect to $f$ .

Proposition 5.1.

Let $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ . For any relatively compact subset $K\subset I$ , there exists a neighbourhood $\mathcal{U}_{K}$ of $K$ in $V$ such that $h_{1}$ and $h_{2}$ are mutually bounded on $\mathcal{U}_{K}\setminus\varpi^{-1}(0)$ .

5.1.2. Maximal non-positive but non-special intervals

Suppose that there exists an interval $I$ in $\varpi^{-1}(0)\cap V$ satisfying the following conditions.

•

$I$ is maximally non-positive but not special with respect to $f$ .
•

$\overline{I}\subset V$ .

Theorem 5.2.

Let $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ . There exists a neighbourhood $\mathcal{U}$ of $\overline{I}$ in $V$ such that $h_{1}$ and $h_{2}$ are mutually bounded on $\mathcal{U}\setminus\varpi^{-1}(0)$ .

5.1.3. Special intervals

Suppose that there exists an interval $I$ in $\varpi^{-1}(0)\cap V$ satisfying the following conditions.

•

$\overline{I}\subset V$ .
•

$I$ is special with respect to $f$ .

There exist a non-zero complex number $\alpha$ and $\rho>0$ such that $\deg(\mathfrak{a}(f,Q)-\alpha z^{-\rho})<\rho$ for any $Q\in I\setminus\mathcal{Z}(f)$ . Let $\mathcal{P}$ be the set of the tuples ${\boldsymbol{a}}=(a_{1},\ldots,a_{r})\in{\mathbb{R}}^{r}$ such that

a_{1}\geq a_{2}\geq\cdots\geq a_{r}\geq a_{1}-1,\quad\quad\sum a_{i}=0.

Theorem 5.3.

•

For any $h\in\mathop{\rm Harm}\nolimits(q)$ , there exist ${\boldsymbol{a}}_{I}(h)=(a_{I,i}(h))\in\mathcal{P}$ and $\epsilon>0$ such that the following holds on $\bigl{\{}|\arg(\alpha z^{-\rho})-\pi|<(1-\delta)\pi/2\bigr{\}}$ for any $\delta>0$ :

\log\bigl{|}(dz)^{(r+1-2i)/2}\bigr{|}_{h}+a_{I,i}(h)\mathop{\rm Re}\nolimits\bigl{(}\alpha z^{-\rho}\bigr{)}=O\bigl{(}|z|^{-\rho+\epsilon}\bigr{)}.

The tuple ${\boldsymbol{a}}_{I}(h)$ is uniquely determined by the condition.

•

For $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ , there exists a neighbourhood $\mathcal{U}$ of $\overline{I}$ in $V$ such that $h_{1}$ and $h_{2}$ are mutually bounded on $\mathcal{U}\setminus\varpi^{-1}(0)$ .
•

For any ${\boldsymbol{a}}\in\mathcal{P}$ , there exists $h\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}$ .

We shall also prove the following auxiliary statement.

Proposition 5.4.

There exist relatively compact open neighbourhoods $\mathcal{U}_{I,i}$ $(i=1,2)$ of $I$ in $V$ , an ${\mathbb{R}}_{\geq 0}$ -valued harmonic function $\phi_{I}$ on $\mathcal{U}_{I,1}\setminus\varpi^{-1}(0)$ , and a smooth exhaustive family $\{K_{i}\}$ of $V\setminus\varpi^{-1}(0)$ such that the following holds.

•

$\mathcal{U}_{I,2}\subset\mathcal{U}_{I,1}$ .
•

$\mathcal{U}_{I,i}\cap\varpi^{-1}(0)=I$ holds, and $\partial(\mathcal{U}_{I,i}\setminus\varpi^{-1}(0))$ is smooth in $V\setminus\varpi^{-1}(0)$ .
•

There exists $\epsilon>0$ such that $\phi_{I}=O\bigl{(}|z|^{-\rho+\epsilon}\bigr{)}$ on $\mathcal{U}_{I,1}\setminus\varpi^{-1}(0)$ .
•

Take $h\in\mathop{\rm Harm}\nolimits(q)$ and $h_{i}\in\mathop{\rm Harm}\nolimits(q_{|K_{i}})$ such that $h_{i|\mathcal{U}_{I,1}\cap\partial K_{i}}=h_{|\mathcal{U}_{I,1}\cap\partial K_{i}}$ . Let $s_{i}$ be the automorphism of $\mathbb{K}_{\mathcal{U}_{I,1}\cap K_{i},r}$ determined by $h_{i|\mathcal{U}_{I,1}\cap K_{i}}=h_{|\mathcal{U}_{I,1}\cap K_{i}}\cdot s_{i}$ . Then, we obtain

$\log\mathop{\rm Tr}\nolimits(s_{i})\leq\phi_{I}$

on $\mathcal{U}_{I,2}\cap K_{i}$ .

5.2. Some estimates on sectors around infinity

5.2.1. Positive and negative cases

For any $R>0$ and $0<L<\pi$ , we set $W(R,L):=\bigl{\{}w\in{\mathbb{C}}\,\big{|}\,|w|>R,\,\,|\arg(w)|<L\bigr{\}}$ . Let $f$ be a holomorphic function on a sector $W(R,L)$ . We set $q=f\,(dw)^{r}$ . Let $Z(f)$ denote the set of the zeroes of $f$ . Let $\rho$ be a positive number such that $\rho<\min\{1,\pi/2L\}$ . Let $\alpha$ be a non-zero complex number. Let $0<\kappa_{1}<\cdots<\kappa_{m}<\rho$ and $\alpha_{i}\in{\mathbb{C}}^{\ast}$ $(i=1,\ldots,m)$ . We set

\mathfrak{a}:=\alpha w^{\rho}+\sum_{i=1}^{m}\alpha_{i}w^{\kappa_{i}}.

Let $a\in{\mathbb{R}}$ and $n\in{\mathbb{Z}}_{\geq 0}$ . We impose the following condition on $f$ in this subsection.

Condition 5.5.

There exists $C_{0}>0$ such that the following holds on $W(R,L)$ :

e^{-\mathop{\rm Re}\nolimits\mathfrak{a}(w)}|w|^{-a}|\log w|^{-n}|f|\leq C_{0}.

(60)

Moreover, for any $L_{2}<L_{1}<L$ , there exist $R_{2}>R_{1}>R$ and subsets $\mathcal{C}_{1}\subset\mathcal{C}_{2}\subset W(R,L)$ such that the following holds.

•

Let $\mathcal{D}$ be any connected component of $\mathcal{C}_{2}$ such that

$\mathcal{D}\cap W(R_{2},L_{2})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $W(R_{1},L_{1})$ .
•

For any $w_{0}\in W(R_{1},L_{1})\setminus\mathcal{C}_{2}$ , $\{|w-w_{0}|<1\}$ is contained in $W(R,L)\setminus\mathcal{C}_{1}$ .
•

There exists $C_{1}>0$ such that the following holds on $W(R,L)\setminus\mathcal{C}_{1}$ :

$e^{-\mathop{\rm Re}\nolimits\mathfrak{a}(w)}|w|^{-a}|\log w|^{-n}|f|\geq C_{1}.$ (61)

Lemma 5.6.

Suppose that $\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\rho\theta})>0$ for $|\theta|\leq L$ . Then, any $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ are mutually bounded on $W(R^{\prime},L^{\prime})$ for any $R^{\prime}>R$ and $0<L^{\prime}<L$ .

Proof Let $s$ be the automorphism of $\mathbb{K}_{W(R,L),r}$ determined by $h_{2}=h_{1}\cdot s$ . We set $L_{2}=L^{\prime}$ . We take $L_{1}$ such that $L_{2}<L_{1}<L$ . We take $R_{2}>R_{1}>R$ and $\mathcal{C}_{1}\subset\mathcal{C}_{2}$ as in Condition 5.5. Let $w_{0}\in W(R_{1},L_{1})\setminus\mathcal{C}_{2}$ . On $\{|w-w_{0}|<1\}$ , the estimates (60) and (61) hold, and we may apply Corollary 3.10. Hence, we obtain the boundedness of $\mathop{\rm Tr}\nolimits(s)$ on $W(R_{1},L_{1})\setminus\mathcal{C}_{2}$ . Let $\mathcal{D}$ be a connected component of $\mathcal{C}_{2}$ with $\mathcal{D}\cap W(R_{2},L_{2})\neq\emptyset$ . Then, it is relatively compact in $W(R_{1},L_{1})$ . Recall that $\mathop{\rm Tr}\nolimits(s)$ is subharmonic (see §2.4). By the maximum principle, we obtain that $\sup_{\mathcal{D}}\mathop{\rm Tr}\nolimits(s)\leq\sup_{W(R_{1},L_{1})\setminus\mathcal{C}_{2}}\mathop{\rm Tr}\nolimits(s)$ . Hence, we obtain the boundedness of $\mathop{\rm Tr}\nolimits(s)$ on $W(R_{2},L_{2})$ . For any $0<R^{\prime}<R_{2}$ , $W(R_{2},L_{2})\setminus W(R^{\prime},L_{2})$ is relatively compact in $W(R,L)$ . Hence, we obtain the boundedness of $\mathop{\rm Tr}\nolimits(s)$ on $W(R^{\prime},L_{2})$ . ∎

Lemma 5.7.

Suppose that $\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\rho\theta})<0$ for $|\theta|\leq L$ . Then, for any $h\in\mathop{\rm Harm}\nolimits(q)$ , and for any $R^{\prime}>R$ and $0<L^{\prime}<L$ , there exists $C>0$ such that the following holds on $W(R^{\prime},L^{\prime})$ :

\Bigl{|}\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}\Bigr{|}\leq C\bigl{|}\mathop{\rm Re}\nolimits(\alpha w^{\rho})\bigr{|}=-C\mathop{\rm Re}\nolimits(\alpha w^{\rho}).

Proof Let $\theta(q)$ denote the Higgs field of $\mathbb{K}_{W(R,L),r}$ associated with $q$ . Take $L^{\prime}<L^{\prime\prime}<L$ . There exist $\epsilon>0$ , $R_{10}>0$ and the map $\Psi:W(R_{10},(1+\epsilon)\pi/2)\longrightarrow W(R,L)$ defined by $\Psi(\zeta)=\zeta^{2L^{\prime\prime}/\pi}$ .

For $\zeta_{1}\in W(2R_{10},\pi/2)$ , we may assume that there exists $\{|\zeta-\zeta_{1}|<1\}\subset\Psi^{-1}(W(R,L))$ . By Proposition 3.7, there exists $C_{10}>0$ , which is independent of $\zeta_{1}$ , such that $|\Psi^{-1}(\theta(q))|_{\Psi^{-1}(h)}\leq C_{10}$ on $\{|\zeta-\zeta_{1}|<1/2\}$ . Hence, we obtain $|\Psi^{-1}(\theta(q))|_{\Psi^{-1}(h)}\leq C_{10}$ on $W(2R_{10},\pi/2)$ .

For $\zeta_{1}\in W(R_{10},\pi/2)$ such that $\mathop{\rm Re}\nolimits(\zeta_{1})>2R_{10}$ , $\bigl{\{}|\zeta-\zeta_{1}|<\mathop{\rm Re}\nolimits(\zeta_{1})-R_{10}\bigr{\}}$ is contained in $W(R_{10},\pi/2)$ . There exists the isomorphism $\Phi_{\zeta_{1}}:\{|\eta|<1\}\simeq\bigl{\{}|\zeta-\zeta_{1}|<\mathop{\rm Re}\nolimits(\zeta_{1})-R_{10}\bigr{\}}$ given by $\Phi_{\zeta_{1}}(\eta)=\zeta_{1}+(\mathop{\rm Re}\nolimits\zeta_{1}-R_{10})\eta$ . By applying Proposition 3.7 to $\Phi_{\zeta_{1}}^{\ast}\Psi^{\ast}(\mathbb{K}_{X(R,L),r},\theta(q),h)$ , we obtain that there exists $C_{11}>0$ , which is independent of $\zeta_{1}$ , such that $\bigl{|}\Phi_{\zeta_{1}}^{\ast}\Psi^{\ast}(\theta(q))\bigr{|}_{\Phi_{\zeta_{1}}^{\ast}\Psi^{\ast}(h)}\leq C_{11}$ on $\{|\eta|<1/2\}$ . Because $\Phi_{\zeta_{1}}^{\ast}(d\zeta)=(\mathop{\rm Re}\nolimits\zeta_{1}-R_{10})d\eta$ , we obtain

\Psi^{\ast}(\theta(q))=O\bigl{(}(|\mathop{\rm Re}\nolimits(\zeta)|+1)^{-1}\bigr{)}\,d\zeta

with respect to $\Psi^{-1}(h)$ on $W(R_{10},\pi/2)$ . Recall that $R(\Psi^{\ast}(h))$ denotes the curvature of the Chern connection of $\Psi^{\ast}(\mathbb{K}_{W(R,L),r},h)$ . The Hitchin equation implies $R(\Psi^{\ast}(h))=O\bigl{(}(|\mathop{\rm Re}\nolimits(\zeta)|+1)^{-2}\bigr{)}\,d\zeta\,d\overline{\zeta}$ on $W(R_{10},\pi/2)$ . It implies the following estimates with respect to $\Psi^{-1}(h)$ on $W(R_{10},(1-\kappa)\pi/2)$ for any $\kappa>0$ :

\Psi^{\ast}(\theta(q))=O(|\zeta|^{-1})\,d\zeta,\quad R(\Psi^{\ast}(h))=O(|\zeta|^{-2})\,d\zeta\,d\overline{\zeta}.

Take $L^{\prime}<L^{\prime\prime\prime}<L^{\prime\prime}$ . Recall that $R(h)$ denotes the curvature of the Chern connection of $(\mathbb{K}_{W(R,L),r},h)$ . By the previous consideration, and by $w=\zeta^{2L^{\prime\prime}/\pi}$ , we obtain the estimates $\theta(q)=O(|w|^{-1})\,dw$ and $R(h)=O(|w|^{-2})\,dw\,d\overline{w}$ with respect to $h$ on $W(R,L^{\prime\prime\prime})$ .

We set $L_{2}:=L^{\prime}$ and take $L_{2}<L_{1}<L^{\prime\prime\prime}$ . Take $R_{2}>R_{1}>R$ and $\mathcal{C}_{1}\subset\mathcal{C}_{2}$ as in Condition 5.5. Note that $\theta(q)=O(|w|^{-1})\,dw$ with respect to $\Psi^{-1}(h)$ on $W(R_{1},L_{1})$ . At any point of $W(R_{1},L_{1})\setminus\mathcal{C}_{1}$ , the estimate (61) holds. By Corollary 3.5, there exists $C_{12}>0$ such that the following holds for $i=1,\ldots,r$ on $W(R_{1},L_{1})\setminus\mathcal{C}_{1}$ :

\Bigl{|}\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}\Bigr{|}\leq C_{12}\bigl{|}\mathop{\rm Re}\nolimits(\alpha w^{\rho})\bigr{|}.

Because $R(h)=O(|w|^{-2})dw\,d\overline{w}$ on $W(R_{1},L^{\prime\prime\prime})$ , we obtain the following estimate on $W(R,L^{\prime\prime\prime})$ :

\partial_{w}\partial_{\overline{w}}\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}=O(|w|^{-2}).

Hence, by Lemma 4.5, we can prove that there exist functions $\beta_{i}$ on $W(R_{1},L_{1})$ such that (i) the functions $\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}-\beta_{i}$ are harmonic, (ii) $|\beta_{i}|=O\bigl{(}\log(|w|+1)\bigr{)}$ . There exists $C_{13}>0$ such that the following holds on $W(R_{1},L_{1})\setminus\mathcal{C}_{1}$ :

-C_{13}\bigl{|}\mathop{\rm Re}\nolimits(\alpha w^{\rho})\bigr{|}\leq\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}-\beta_{i}\leq C_{13}\bigl{|}\mathop{\rm Re}\nolimits(\alpha w^{\rho})\bigr{|}.

(62)

Let $\mathcal{D}$ be any connected component of $\mathcal{C}_{2}$ such that $\mathcal{D}\cap W(R_{2},L_{2})\neq\emptyset$ . Because it is relatively compact in $W(R_{1},L_{1})$ , the inequalities (62) hold on $\mathcal{D}$ . Hence, (62) holds on $W(R_{2},L_{2})$ . Then, we can deduce the claim of the lemma easily. ∎

5.2.2. Neutral case

We set

\widetilde{W}(R,L):=\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,\mathop{\rm Re}\nolimits(\zeta)>\log R,\,\,|\mathop{\rm Im}\nolimits(\zeta)|<L\bigr{\}}.

Let $\Phi_{1}:{\mathbb{C}}\longrightarrow{\mathbb{C}}^{\ast}$ be given by $\Phi_{1}(\zeta)=e^{\zeta}$ . It induces $\widetilde{W}(R,L)\simeq W(R,L)$ . Let $a\in{\mathbb{R}}$ and $n\in{\mathbb{Z}}_{\geq 0}$ . We impose the following condition to $f$ .

Condition 5.8.

For any $L_{2}<L_{1}<L$ , there exist $R_{2}>R_{1}>R$ and subsets $\mathcal{B}_{1}\subset\mathcal{B}_{2}\subset\widetilde{W}(R,L)$ such that the following holds.

•

Let $\mathcal{D}$ be any connected component of $\mathcal{B}_{2}$ such that

$\mathcal{D}\cap\widetilde{W}(R_{2},L_{2})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $\widetilde{W}(R_{1},L_{1})$ .
•

For any $\zeta_{0}\in\widetilde{W}(R_{1},L_{1})\setminus\mathcal{B}_{2}$ , we obtain $\{|\zeta-\zeta_{0}|<1\}\subset\widetilde{W}(R,L)\setminus\mathcal{B}_{1}$ .
•

There exists $C>1$ such that the following holds on $\widetilde{W}(R,L)\setminus\mathcal{B}_{1}$ :

$C^{-1}\leq e^{-a\mathop{\rm Re}\nolimits(\zeta)}|\zeta|^{-n}|\Phi_{1}^{\ast}(f)(\zeta)|\leq C.$ (63)

Lemma 5.9.

If Condition 5.8 is satisfied, the following holds.

•

If $a\geq-r$ , any $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ are mutually bounded on $W(R^{\prime},L^{\prime})$ for any $R^{\prime}>R$ and $0<L^{\prime}<L$ .
•

Suppose $a<-r$ . Then, for any $h\in\mathop{\rm Harm}\nolimits(q)$ , and for any $R^{\prime}>R$ and $0<L^{\prime}<L$ , there exists $C>0$ such that the following holds for $i=1,\ldots,r$ on $W(R^{\prime},L^{\prime})$ :

$\Bigl{|}\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}\Bigr{|}\leq C\log|w|.$ (64)

Proof Let us study the first claim. We note that $\Phi_{1}^{\ast}\bigl{(}f\cdot(dw)^{r}\bigr{)}=\Phi_{1}^{\ast}(f)\cdot e^{r\zeta}(d\zeta)^{r}$ . The inequalities (63) is rewritten as follows:

C^{-1}\leq e^{-(a+r)\mathop{\rm Re}\nolimits(\zeta)}|\zeta|^{-n}\bigl{|}\Phi_{1}^{\ast}(f)(\zeta)\cdot e^{r\zeta}\bigr{|}\leq C.

For $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ , let $s$ be the automorphism of $\mathbb{K}_{\widetilde{W}(R,L),r}$ determined by $h_{2}=h_{1}\cdot s$ . We set $L_{2}:=L^{\prime}$ , and take $L_{2}<L_{1}<L$ . Let $R_{2}>R_{1}>R$ and $\mathcal{B}_{1}\subset\mathcal{B}_{2}$ as in Condition 5.8.

•

If $(a+r,n)\in({\mathbb{R}}_{\geq 0}\times{\mathbb{Z}}_{\geq 0})\setminus\{(0,0)\}$ , we obtain that $\mathop{\rm Tr}\nolimits(s)$ is bounded on $\widetilde{W}(R_{1},L_{1})\setminus\mathcal{B}_{2}$ by Corollary 3.10.
•

If $(a+r,n)=(0,0)$ , we obtain the boundedness of $\mathop{\rm Tr}\nolimits(s)$ on $\widetilde{W}(R_{1},L_{1})\setminus\mathcal{B}_{2}$ from Corollary 3.8.

By applying the argument in the proof of Lemma 5.6, we obtain that $\mathop{\rm Tr}\nolimits(s)$ is bounded on $\widetilde{W}(R_{2},L_{2})$ . Then, we obtain the first claim immediately.

Let us study the second claim. Let $h\in\mathop{\rm Harm}\nolimits(q)$ . By using the argument in the proof of Lemma 5.7, we can prove that there exists $C_{1}>0$ such that the following holds on $\widetilde{W}(R^{\prime},L^{\prime})$ :

\Bigl{|}\log\bigl{|}(d\zeta)^{(r+1-2i)/2}\bigr{|}_{\Phi_{1}^{\ast}(h)}\Bigr{|}\leq C_{1}\bigl{(}|\mathop{\rm Re}\nolimits(\zeta)|+1\bigr{)}

(65)

Because $\Phi_{1}^{\ast}(dw/w)=d\zeta$ , we obtain (64) from (65). ∎

5.2.3. Estimates on sectors around infinity in the multiple case

Let us continue to use the notation in §5.2.1.

Lemma 5.10.

Suppose that $f$ satisfies the following conditions.

•

There exist $C_{0}>0$ and $\kappa_{0}>0$ such that $|f|\leq C_{0}\exp(\kappa_{0}|w|^{\rho})$ on $W(R,L)$ .
•

There exist $C_{1}>0$ , $\kappa_{1}>0$ , and a subset $Z_{1}\subset{\mathbb{R}}_{>0}$ with $\int_{Z_{1}}dt/t<\infty$ , such that $|f|\geq C_{1}\exp(-\kappa_{1}|w|^{\rho})$ on $\bigl{\{}w\in W(R,L)\,\big{|}\,|w|\not\in Z_{1}\bigr{\}}$ .

Let $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits(q)$ . Let $s$ be the automorphism of $\mathbb{K}_{W(R,L),r}$ determined by $h_{2}=h_{1}s$ . Then, for any $R^{\prime}>R$ and $0<L^{\prime}<L$ , there exist $C>0$ such that $\bigl{|}\log\mathop{\rm Tr}\nolimits(s)\bigr{|}\leq C|w|^{\rho}$ on $\bigl{\{}w\in W(R^{\prime},L^{\prime})\,\big{|}\,|w|\not\in Z_{1}\bigr{\}}$ .

Proof By Proposition 3.7, there exists a constant $C_{10}>0$ and $\kappa_{10}>0$ such that the following holds on $W(R^{\prime},L^{\prime})$ :

|\theta(q)|_{h_{j}}\leq C_{10}\exp(\kappa_{10}|w|^{\rho}).

By Corollary 3.6, there exist $C_{11}>0$ and $\kappa_{11}>0$ such that the following holds on $\bigl{\{}w\in W(R^{\prime},L^{\prime})\,\big{|}\,|w|\not\in Z_{1}\bigr{\}}$ :

|s|_{h_{1}}\leq C_{11}\exp(\kappa_{11}|w|^{\rho}).

Thus, we obtain the claim of the lemma. ∎

5.3. Proof of Proposition 5.1 and Theorem 5.2

5.3.1. Proof of Proposition 5.1

It is enough to prove the following claim for any $Q\in I$ .

•

There exists a relatively compact neighbourhood $\mathcal{U}$ of $Q$ in $V$ such that $h_{1}$ and $h_{2}$ are mutually bounded on $\mathcal{U}\setminus\varpi^{-1}(0)$ .

If $Q\in I\setminus\mathcal{Z}(f)$ , the claim follows from Lemma 5.6. Let us study the case where $Q\in I\cap\mathcal{Z}(f)$ . We may assume that $I\cap\mathcal{Z}(f)=\{Q\}$ . Let $s$ be determined by $h_{2}=h_{1}\cdot s$ . We have already proved that for any $Q^{\prime}\in I\setminus\{Q\}$ there exists a neighbourhood $\mathcal{U}_{Q^{\prime}}$ in $V$ such that $s$ is bounded on $\mathcal{U}_{Q^{\prime}}\setminus\varpi^{-1}(0)$ . By Lemma 4.39, Lemma 4.40 and Lemma 5.10, there exist $N>0$ , a neighbourhood of $\mathcal{U}$ of $Q$ in $V$ , and a subset $Z_{1}\subset{\mathbb{R}}_{>0}$ with $\int_{Z_{1}}dt/t<\infty$ such that $\log\mathop{\rm Tr}\nolimits(s)=O(|z|^{-N})$ on $\mathcal{U}\setminus\bigl{(}\varpi^{-1}(0)\cup(Z_{1}\times S^{1})\bigr{)}$ . Then, we obtain the boundedness of $\log\mathop{\rm Tr}\nolimits(s)$ on $\mathcal{U}\setminus\varpi^{-1}(0)$ by Corollary 4.3. ∎

5.3.2. Proof of Theorem 5.2

Let $Q_{1},Q_{2}$ denote the points of $\partial I$ . Let $I_{i}$ be a neighbourhood of $Q_{i}$ in $\varpi^{-1}(0)\cap V$ such that $I_{i}\cap\mathcal{Z}(f)=\{Q_{i}\}$ . Then, $f$ is simply positive at each point of $I_{i}\setminus\overline{I}$ .

Let us study the case where $I$ is negative with respect to $f$ . Then, there exists $\rho>0$ such that $\deg(\mathfrak{a}(f,Q))=\rho$ for any $Q\in I\setminus\mathcal{Z}(f)$ . Because $I$ is not special with respect to $f$ , the length of $I$ is strictly less than $\pi/\rho$ .

Let $s$ be the automorphism determined by $h_{2}=h_{1}\cdot s$ . According to Proposition 5.1, $\mathop{\rm Tr}\nolimits(s)$ is bounded around any point of $(I_{1}\cup I_{2})\setminus\overline{I}$ . According to Lemma 5.7, we obtain $\log\mathop{\rm Tr}\nolimits(s)=O\Bigl{(}|z|^{-\rho}\Bigr{)}$ around any point of $I\setminus\mathcal{Z}(f)$ . Moreover, according to Lemma 5.10, there exist $N>0$ , a neighbourhood $\mathcal{U}_{0}$ of $\mathcal{Z}(f)\cap\overline{I}$ in $V$ , and a subset $Z_{1}\subset{\mathbb{R}}_{>0}$ with $\int_{Z_{1}}dt/t<\infty$ , such that $\log\mathop{\rm Tr}\nolimits(s)=O\Bigl{(}|z|^{-N}\Bigr{)}$ on $\mathcal{U}\setminus\bigl{(}\varpi^{-1}(0)\cup(Z_{1}\times S^{1})\bigr{)}$ . By Corollary 4.3, we obtain that $\log\mathop{\rm Tr}\nolimits(s)=O(|z|^{-\rho})$ on $\mathcal{U}\setminus\varpi^{-1}(0)$ . Because the length of $I$ is strictly smaller than $\pi/\rho$ , the Phragmén-Lindelöf theorem (see Corollary 4.2) implies that $\log\mathop{\rm Tr}\nolimits(s)$ is bounded. Thus, we obtain the claim of the proposition in the case where $I$ is negative with respect to $f$ .

We can prove the claim by a similar and easier argument in the case where $I$ is neutral with respect to $f$ by using Lemma 5.9 instead of Lemma 5.7. ∎

5.4. Outline of the proof of Theorem 5.3 and Proposition 5.4

5.4.1. Setting in a simple case

Let $\varpi_{\infty}:\widetilde{\mathbb{P}}^{1}_{\infty}\longrightarrow\mathbb{P}^{1}$ denote the oriented real blow up at $\infty$ . We identify $\varpi_{\infty}^{-1}(\infty)\simeq S^{1}$ by the polar decomposition $w=|w|e^{\sqrt{-1}\theta}$ .

For $R>0$ and $\delta>0$ , we set

X(R,\delta):=\bigl{\{}w\in{\mathbb{C}}\,\big{|}\,|w|>R,\,|\arg(w)-\pi/2|<\pi/2+\delta\bigr{\}}.

We regard $X(R,\delta)$ as an open subset of $\widetilde{\mathbb{P}}^{1}_{\infty}$ . Let $\overline{X}(R,\delta)$ denote the closure of $X(R,\delta)$ in $\widetilde{\mathbb{P}}^{1}_{\infty}$ .

For the proof of Theorem 5.3 and Proposition 5.4, we introduce the coordinate $w$ determined by $\alpha z^{-\rho}=\sqrt{-1}w$ , and we study harmonic metrics of the Higgs bundle associated with $q=f(dw)^{r}$ on $X(R,\delta)$ such that $\deg(\mathfrak{a}(f,\theta)-\sqrt{-1}w)<\deg(\mathfrak{a}(f,\theta))$ for any $0<\theta<\pi$ .

In the following of this subsection, $C_{i}$ will denote positive constants.

As an outline of the proof of Theorem 5.3 and Proposition 5.4, we explain rather detail of our arguments by assuming that $f$ satisfies the following stronger condition.

•

There exist $C_{i}>0$ $(i=1,2,3)$ such that the following inequalities are satisfied on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

$|f|\leq C_{1}e^{-\mathop{\rm Im}\nolimits(w)}.$ (66)

$|f|\geq C_{2}e^{-\mathop{\rm Im}\nolimits(w)-C_{3}|w|^{1-\epsilon}}.$ (67)
•

There exist $C_{i}>0$ $(i=4,5,6)$ such that

$C_{4}e^{-C_{4}|w|^{C_{4}}}\leq|f|\leq C_{5}e^{C_{5}|w|^{C_{5}}}$ (68)

on $\bigl{\{}|\arg(w)|<C_{6}\bigr{\}}\cup\bigl{\{}|\arg(w)-\pi|<C_{6}\bigr{\}}$ .

Note that the interval $\{\infty\}\times]0,\pi[$ is special with respect to $f$ , and hence $f$ is simply positive at any points of $\{\infty\}\times\bigl{(}]-\delta,0[\cup]\pi,\pi+\delta[\bigr{)}$ .

5.4.2. Parabolic structure

Let us explain an outline of the proof of the first claim of Theorem 5.3 under the simplified setting in §5.4.1. (We shall study a general case in §5.6.1.)

Let $h\in\mathop{\rm Harm}\nolimits(q)$ . For any $w_{1}$ with $\mathop{\rm Im}\nolimits(w_{1})>R$ , there exists an isomorphism

\Phi_{w_{1}}:\{|\eta|<1\}\simeq\{|w-w_{1}|<\mathop{\rm Im}\nolimits(w_{1})-R\}

defined by $\Phi_{w_{1}}(\eta)=w_{1}+(\mathop{\rm Im}\nolimits(w_{1})-R)\eta$ . Note that $\Phi_{w_{1}}^{\ast}(f(dw)^{r})=\Phi_{w_{1}}^{\ast}(f)\cdot(\mathop{\rm Im}\nolimits(w_{1})-R)^{r}\,(d\eta)^{r}$ . By (66), there exists $C^{(1)}_{1}>0$ , which is independent of $w_{1}$ such that the following holds on $\{|\eta|<1\}$ :

|\Phi_{w_{1}}^{\ast}(f)|(\mathop{\rm Im}\nolimits(w_{1})-R)^{r}\leq C^{(1)}_{1}.

Hence, by Proposition 3.7, there exists $C^{(1)}_{2}>0$ , which is independent of $w_{1}$ , such that the following holds on $\{|\eta|<1/2\}$ :

|\Phi_{w_{1}}^{\ast}(\theta(q))|_{\Phi_{w_{1}}^{\ast}(h)}\leq C^{(1)}_{2}.

Because $\Phi_{w_{1}}^{\ast}(dw)=(\mathop{\rm Im}\nolimits(w_{1})-R)\,d\eta$ , there exists $C^{(1)}_{3}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

|\theta(q)|_{h}\leq C^{(1)}_{3}(\mathop{\rm Im}\nolimits w-R)^{-1}.

(69)

By the Hitchin equation, there exists $C^{(1)}_{4}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

|R(h)|_{h}\leq C^{(1)}_{4}(\mathop{\rm Im}\nolimits w-R)^{-2}.

(70)

By (70), there exists $C^{(1)}_{5}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

\Bigl{|}\partial_{z}\partial_{\overline{z}}\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}\Bigr{|}\leq C^{(1)}_{5}(\mathop{\rm Im}\nolimits w-R)^{-2}.

(71)

By (67), (69) and Corollary 3.5, there exists $C^{(1)}_{6}$ such that

\Bigl{|}\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}\Bigr{|}\leq C^{(1)}_{6}(\mathop{\rm Im}\nolimits(w)+|w|^{1-\epsilon}).

(72)

By (71), (72) and the Nevanlinna formula (Proposition 4.4), there exists $a_{i}(h)$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

\log|(dw)^{(r+1-2i)/2}|_{h}-a_{i}(h)\mathop{\rm Im}\nolimits(w)=O(|w|^{1-\epsilon}).

(73)

Because of (69) and $\theta(q)(dw)^{(r+1-2i)/2}=(dw)^{(r+1-2(i+1))/2}$ , we obtain $a_{i}(h)\geq a_{i+1}(h)$ . By (67), (69), and the relation $\theta(q)(dw)^{(-r+1)/2}=f(dw)^{(r-1)/2}$ , we also obtain $a_{r}(h)\geq a_{1}(h)-1$ . In this way, we obtain the parabolic structure of $h$ .

5.4.3. Mutually boundedness

Let us explain an outline of the proof of the second claim of Theorem 5.3 under the simplified setting in §5.4.1. (We shall study a general case in §5.6.2.)

Suppose that $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ satisfy ${\boldsymbol{a}}(h_{1})={\boldsymbol{a}}(h_{2})$ . Let $s$ be the automorphism of $\mathbb{K}_{X(R,\delta),r}$ determined by $h_{2}=h_{1}\cdot s$ . By (21), $\log\mathop{\rm Tr}\nolimits(s)$ is subharmonic.

Note that $f$ is simply positive at any points of $\{\infty\}\times\bigl{(}]-\delta,0[\cup]\pi,\pi+\delta[\bigr{)}$ . Hence, by Proposition 5.1, $h_{1}$ and $h_{2}$ are mutually bounded on $X(R_{1},\delta_{1})\setminus X(R_{1},\delta_{2})$ for any $R_{1}>R$ and $0<\delta_{2}<\delta_{1}<\delta$ .

By the assumption ${\boldsymbol{a}}(h_{1})={\boldsymbol{a}}(h_{2})$ , and by (73), we obtain

\log\mathop{\rm Tr}\nolimits(s)=O(|w|^{1-\epsilon})

on $\{\mathop{\rm Im}\nolimits(w)>R\}$ . By (68) and Proposition 3.7, there exists $C^{(2)}_{1}$ such that the following holds on $\{|\arg(w)|<C_{6}/2\}\cup\{|\arg(w)-\pi|<C_{6}/2\}$ :

|\theta(q)|_{h_{j}}\leq C^{(2)}_{1}e^{C^{(2)}_{1}|w|^{C_{1}^{(2)}}}\quad(j=1,2).

(74)

By (68), (74) and Corollary 3.6, there exists $C^{(2)}_{2}$ such that the following holds on $\{|\arg(w)|<C_{6}/2\}\cup\{|\arg(w)-\pi|<C_{6}/2\}$ :

\log\mathop{\rm Tr}\nolimits(s)\leq C^{(2)}_{2}\bigl{(}1+|w|^{C_{2}^{(2)}}\bigr{)}.

(75)

By using Phragmén-Lindelöf theorem (Corollary 4.3) on small sectors around $\arg(w)=0$ and $\arg(w)=\pi$ , we obtain that there exists $C_{3}^{(2)}$ such that the following holds on $X(R_{1},\delta_{1})$ :

\log\mathop{\rm Tr}\nolimits(s)\leq C_{3}^{(2)}(1+|w|^{1-\epsilon}).

By using Phragmén-Lindelöf theorem (Corollary 4.2) again, we obtain that $\log\mathop{\rm Tr}\nolimits(s)$ is bounded on $X(R_{1},\delta_{3})$ if $\delta_{3}>0$ is sufficiently small. Then, we obtain that $\log\mathop{\rm Tr}\nolimits(s)$ is bounded on $X(R_{1},\delta_{4})$ for any $0<\delta_{4}<\delta$ , i.e. $h_{1}$ and $h_{2}$ are mutually bounded on $X(R_{1},\delta_{4})$ .

5.4.4. Auxiliary metrics

We introduce auxiliary metrics under the simplified setting in §5.4.1. (We shall study a general case in §5.6.3.)

We set $q_{0}=e^{\sqrt{-1}w}(dw)^{r}$ on ${\mathbb{C}}$ . For any ${\boldsymbol{a}}\in\mathcal{P}$ , there exists $h_{0}\in\mathop{\rm Harm}\nolimits(q_{0})$ such that ${\boldsymbol{a}}(h_{0})={\boldsymbol{a}}$ . (See Proposition 3.25.) There exists $C^{(3)}_{1}$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

|\theta(q_{0})|_{h_{0}}\leq C^{(3)}_{1}\mathop{\rm Im}\nolimits(w)^{-1}.

(76)

Note that

\theta(q)(dw)^{(r+1-2i)/2}=(dw)^{(r+1-2(i+1))/2}\,dw=\theta(q_{0})(dw)^{(r+1-2i)/2}

for $i=1,\ldots,r-1$ . We also note that

\theta(q)(dw)^{(-r+1)/2}=f(dw)^{(r-1)/2}\,dw,

\theta(q_{0})(dw)^{(-r+1)/2}=e^{\sqrt{-1}w}(dw)^{(r-1)/2}dw.

We obtain

|\theta(q)|_{h_{0}}=O(|\theta(q_{0})|_{h_{0}}).

Hence, there exists $C^{(3)}_{2}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

|\theta(q)|_{h_{0}}\leq C^{(3)}_{2}\mathop{\rm Im}\nolimits(w)^{-1}.

(77)

By the Hitchin equation for $(\mathbb{K}_{{\mathbb{C}},r},\theta(q_{0}),h_{0})$ , there exists $C^{(3)}_{3}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

|R(h_{0})|_{h_{0}}\leq C^{(3)}_{3}\mathop{\rm Im}\nolimits(w)^{-2}.

We set $F(h_{0},\theta(q)):=R(h_{0})+[\theta(q),\theta(q)^{\dagger}_{h_{0}}]$ . There exists $C^{(3)}_{4}$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

|F(h_{0},\theta(q))|_{h_{0}}\leq C_{4}^{(3)}\mathop{\rm Im}\nolimits(w)^{-2}.

(78)

5.4.5. Comparison with auxiliary metric

We explain how to obtain an estimate for the difference between a harmonic metric and an auxiliary metric. (We shall study a general case in §5.6.4.)

Let $Y$ be a relatively compact open subset of $X(R,\delta)$ such that $\partial Y$ is smooth, and that

Y\cap\{2R-1<\mathop{\rm Im}\nolimits w<4R+1\}=\\ \{2R-1<\mathop{\rm Im}\nolimits w<4R+1,\,|\mathop{\rm Re}\nolimits(w)|<a_{Y}\}

(79)

for some $a_{Y}>2$ . According to Proposition 2.1, there exists $h_{Y}\in\mathop{\rm Harm}\nolimits(q_{|Y})$ such that $h_{Y|\partial Y}=h_{0|\partial Y}$ . Let $s_{Y}$ be the automorphism of $\mathbb{K}_{Y,r}$ determined by $h_{Y}=h_{0|Y}s_{Y}$ . In the following $C^{(4)}_{i}$ are positive constants which are independent of $Y$ .

By Proposition 3.7 and (66), there exists $C^{(4)}_{1}$ such that the following holds on $\{2R<\mathop{\rm Im}\nolimits w<4R,\,|\mathop{\rm Re}\nolimits(w)|<a_{Y}-1\}$ :

|\theta(q)|_{h_{Y}}\leq C_{1}^{(4)}.

(80)

Note that $|\theta(q)|_{h_{0}}=|\theta(q)|_{h_{Y}}$ on $\partial Y$ . By Proposition 3.11, (66), and (77), there exists $C^{(4)}_{2}$ , such that the following holds on $\{2R<\mathop{\rm Im}\nolimits w<4R,\,a_{Y}-1<|\mathop{\rm Re}\nolimits(w)|<a_{Y}\}$ :

|\theta(q)|_{h_{Y}}\leq C_{2}^{(4)}.

(81)

By (67), (76), (80), (81) and Corollary 3.6, there exists $C^{(4)}_{3}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)=3R\}\cap Y$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C_{3}^{(4)}|w|^{1-\epsilon}.

(82)

There exists a complex number $\gamma$ such that $\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})>0$ on $\{\mathop{\rm Im}\nolimits(w)>R\}$ . There exists $C^{(4)}_{4}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)=3R\}\cap Y$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C_{4}^{(4)}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon}).

(83)

By (21) and (78), there exists $C^{(4)}_{5}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>R\}$ :

-\partial_{z}\partial_{\overline{z}}\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C^{(4)}_{5}(\mathop{\rm Im}\nolimits w)^{-2}.

Hence, there exists $C^{(4)}_{6}>0$ such that

-\partial_{z}\partial_{\overline{z}}\Bigl{(}\log\mathop{\rm Tr}\nolimits(s_{Y})-C^{(4)}_{3}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})-C^{(4)}_{6}\log(\mathop{\rm Im}\nolimits(w))-\log r\Bigr{)}\leq 0.

(84)

On $Y\cap\{\mathop{\rm Im}\nolimits(w)=3R\}$ , we obtain

\log\mathop{\rm Tr}\nolimits(s_{Y})-C^{(4)}_{3}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})-C^{(4)}_{6}\log(\mathop{\rm Im}\nolimits(w))-\log r\\ \leq\log\mathop{\rm Tr}\nolimits(s_{Y})-C^{(4)}_{3}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})\leq 0.

(85)

On $\partial(Y)\cap\{\mathop{\rm Im}\nolimits(w)\geq 3R\}$ , we obtain

\log\mathop{\rm Tr}\nolimits(s_{Y})-C^{(4)}_{3}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})-C^{(4)}_{6}\log(\mathop{\rm Im}\nolimits(w))-\log r\leq\\ \log\mathop{\rm Tr}\nolimits(s_{Y})-\log r\leq 0.

(86)

Hence, we obtain $\log\mathop{\rm Tr}\nolimits(s_{Y})-C^{(4)}_{3}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})-C^{(4)}_{6}\log(\mathop{\rm Im}\nolimits(w))-\log r\leq 0$ on $Y\cap\{\mathop{\rm Im}\nolimits(w)\geq 3R\}$ . There exists $C^{(4)}_{7}$ such that the following holds on $Y\cap\{\mathop{\rm Im}\nolimits(w)\geq 3R\}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C^{(4)}_{7}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})=:\phi.

(87)

5.4.6. Construction of harmonic metrics

Let us explain an outline of the proof of the third claim of Theorem 5.3 under the simplified setting in §5.4.1. (We shall study a general case in §5.6.5.)

Suppose that $R>10$ . Let ${\boldsymbol{a}}\in\mathcal{P}$ . We take $h_{0}\in\mathop{\rm Harm}\nolimits(q_{0})$ such that ${\boldsymbol{a}}(h_{0})={\boldsymbol{a}}$ . Let $Y_{i}$ be a smooth exhaustive family of $X(R,\delta)$ such that $Y_{i}\cap\{2R-1<\mathop{\rm Im}\nolimits w<4R+1\}=\{2R-1<\mathop{\rm Im}\nolimits w<4R+1,\,\,|\mathop{\rm Re}\nolimits w|<a_{Y_{i}}\}$ for $a_{Y_{i}}>2$ . We obtain $h_{Y_{i}}\in\mathop{\rm Harm}\nolimits(q_{|Y_{i}})$ such that $h_{Y_{i}|\partial Y_{i}}=h_{0|\partial Y_{i}}$ . According to Proposition 3.15, we may assume that the sequence $\{h_{Y_{i}}\}$ is convergent to $h_{\infty}\in\mathop{\rm Harm}\nolimits(q)$ . By (87), we obtain $\log\mathop{\rm Tr}\nolimits(h_{Y_{i}}h_{0}^{-1})\leq\phi$ on $Y_{i}\cap\{\mathop{\rm Im}\nolimits(w)\geq 3R\}$ , independently of $i$ . Hence, we obtain $\log\mathop{\rm Tr}\nolimits(h_{Y_{\infty}}h_{0}^{-1})\leq\phi$ , which implies ${\boldsymbol{a}}(h_{\infty})={\boldsymbol{a}}(h_{0})={\boldsymbol{a}}$ .

5.4.7. Comparison with harmonic metrics

Let us explain an outline of the proof of Proposition 5.4 under the simplified setting in §5.4.1. (We shall study a general case in §5.6.6.)

Suppose that $R>10$ . Let $Y$ be a relatively compact open subset of $X(R,\delta)$ such that $\partial Y$ is smooth and that it satisfies (79). Let $h\in\mathop{\rm Harm}\nolimits(q)$ . Let $h_{Y}\in\mathop{\rm Harm}\nolimits(q_{|Y})$ such that $h_{Y|\partial Y\cap\{\mathop{\rm Im}\nolimits(w)>R\}}=h_{|\partial Y\cap\{\mathop{\rm Im}\nolimits(w)>R\}}$ . We obtain the automorphism $s_{Y}$ determined by $h_{Y}=h_{|Y}\cdot s_{Y}$ . In the following, constants $C^{(5)}_{i}$ are independent of $h$ and $Y$ .

By (66), there exists $C^{(5)}_{1}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)>2R\}$ :

|\theta(q)|_{h}\leq C^{(5)}_{1}\mathop{\rm Im}\nolimits(w)^{-1}.

(88)

Because $h_{Y|\partial Y\cap\{\mathop{\rm Im}\nolimits(w)>R\}}=h_{|\partial Y\cap\{\mathop{\rm Im}\nolimits(w)>R\}}$ , the following holds on $\partial Y\cap\{\mathop{\rm Im}\nolimits(w)>2R\}$ :

|\theta(q)|_{h_{Y}}\leq C^{(5)}_{1}\mathop{\rm Im}\nolimits(w)^{-1}.

(89)

By Proposition 3.7 and (66), there exists $C^{(5)}_{2}$ such that the following holds on $\{2R<\mathop{\rm Im}\nolimits w<4R,\,|\mathop{\rm Re}\nolimits(w)|\leq a_{Y}-1\}$ :

|\theta(q)|_{h_{Y}}\leq C_{2}^{(5)}.

(90)

By Proposition 3.11, (66) and (89), there exists $C^{(5)}_{3}$ such that the following holds on $\{2R<\mathop{\rm Im}\nolimits w<4R,\,a_{Y}-1\leq|\mathop{\rm Re}\nolimits(w)|\leq a_{Y}\}$ :

|\theta(q)|_{h_{Y}}\leq C_{3}^{(5)}.

(91)

By (67), (88), (90), (91), and Corollary 3.6, there exists $C^{(5)}_{4}>0$ such that the following holds on $\{\mathop{\rm Im}\nolimits(w)=3R\}\cap Y$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C_{4}^{(5)}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon}).

(92)

Because $\log\mathop{\rm Tr}\nolimits(s_{Y})$ is subharmonic by (21), we obtain the following on $Y\cap\{\mathop{\rm Im}\nolimits(w)\geq 3R\}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C_{5}^{(5)}\mathop{\rm Re}\nolimits(\gamma w^{1-\epsilon})+\log r.

5.5. Preliminary for the proof of Theorem 5.3 and Proposition 5.4

We prepare some notation and lemmas for the proof of Theorem 5.3 and Proposition 5.4 in §5.6.

5.5.1. Setting

Take $R_{0}>10$ and $0<\delta_{0}<\pi/2$ . Let $f$ be a holomorphic function on $X(R_{0},\delta_{0})$ , which induces a section of $\mathfrak{B}_{\widetilde{\mathbb{P}}^{1}_{\infty}}$ on $\overline{X}(R_{0},\delta_{0})$ . We also assume the following.

•

$\mathcal{Z}(f)\cap\bigl{(}]-\delta_{0},0[\cup]\pi,\pi+\delta_{0}[\bigr{)}=\emptyset$ .
•

For $\theta\in]0,\pi[\setminus\mathcal{Z}(f)$ , $\mathfrak{a}(f,\theta)$ is of the form:

$\mathfrak{a}(f,\theta)=\sqrt{-1}w+\sum_{0<\kappa<1}\mathfrak{a}(f,\theta)_{\kappa}w^{\kappa}.$

Note that $\mathfrak{a}(f,\theta)_{\kappa}$ are constant on each connected component of $]0,\pi[\setminus\mathcal{Z}(f)$ .

For each $\theta\in]-\delta_{0},\pi+\delta_{0}[$ , there exist a finite subset $\mathcal{I}(f,\theta)\subset\bigoplus_{\kappa>0}{\mathbb{C}}w^{\kappa}$ and a sector

W(f,\theta)=\bigl{\{}|w|>R(\theta),|\arg(w)-\theta|<L(\theta)\bigr{\}}

such that $f$ is expressed as $f=\sum_{\mathfrak{a}\in\mathcal{I}(f,\theta)}f_{\theta,\mathfrak{a}}$ on $W(f,\theta)$ , where $f_{\theta,\mathfrak{a}}$ has a single growth order $\mathfrak{a}$ , and $R(\theta)$ and $L(\theta)$ are positive numbers. We assume that the expression $f=\sum_{\mathfrak{a}\in\mathcal{I}(f,\theta)}f_{\theta,\mathfrak{a}}$ is reduced, i.e., for two distinct elements $\mathfrak{a},\mathfrak{b}\in\mathcal{I}(f,\theta)$ , neither $\mathfrak{a}\prec_{\theta}\mathfrak{b}$ nor $\mathfrak{b}\prec_{\theta}\mathfrak{a}$ holds. (See Lemma 4.22.)

Lemma 5.11.

If $0<\theta<\pi$ , we obtain $\deg(\mathfrak{a}-\sqrt{-1}w)<1$ for any $\mathfrak{a}\in\mathcal{I}(f,\theta)$ . For each $\mathfrak{b}=\sum_{\kappa>0}\mathfrak{b}_{\kappa}w^{\kappa}\in\mathcal{I}(f,0)$ , the following holds.

•

We have $\deg(\mathfrak{b})\geq 1$ . If $\deg(\mathfrak{b})>1$ , we obtain $-\sqrt{-1}\mathfrak{b}_{\deg(\mathfrak{b})}>0$ . If $\deg(\mathfrak{b})=1$ , we obtain $-\sqrt{-1}\mathfrak{b}_{\deg(\mathfrak{b})}\geq 1$ .

For each $\mathfrak{c}=\sum_{\kappa>0}\mathfrak{c}_{\kappa}w^{\kappa}\in\mathcal{I}(f,\pi)$ , the following holds.

•

We have $\deg(\mathfrak{c})\geq 1$ . If $\deg(\mathfrak{c})>1$ , we obtain

$-\sqrt{-1}\mathfrak{c}_{\deg(\mathfrak{c})}e^{\pi\sqrt{-1}\deg(\mathfrak{c})}<0.$

If $\deg(\mathfrak{c})=1$ , we obtain $\sqrt{-1}\mathfrak{c}_{\deg(\mathfrak{c})}\leq-1$ .

Proof Let $0<\theta<\pi$ and $\mathfrak{a}\in\mathcal{I}(f,\theta)$ . We have the expression:

\mathfrak{a}=\sum_{0<\kappa\leq\deg(\mathfrak{a})}\mathfrak{a}_{\kappa}w^{\kappa}.

Take $\theta_{-}<\theta<\theta_{+}$ such that $|\theta-\theta_{\pm}|$ is sufficiently small. We have the expressions

\mathfrak{a}(f,\theta_{\pm})=\sqrt{-1}w+\sum_{0<\kappa<1}\mathfrak{a}(f,\theta_{\pm})_{\kappa}w^{\kappa}.

If $\deg(\mathfrak{a})>1$ , we obtain $\mathop{\rm Re}\nolimits(\mathfrak{a}_{\deg(\mathfrak{a})}e^{\sqrt{-1}\theta_{\pm}\deg(\mathfrak{a})})<0$ from $\mathfrak{a}\prec_{\theta_{\pm}}\mathfrak{a}(f,\theta_{\pm})$ . It implies that

\mathop{\rm Re}\nolimits(\mathfrak{a}_{\deg(\mathfrak{a})}e^{\sqrt{-1}\theta\deg(\mathfrak{a})})<0,

and hence we obtain $\mathfrak{a}\prec_{\theta}\mathfrak{a}(f,\theta_{\pm})$ , which contradicts that $\mathfrak{a},\mathfrak{a}(f,\theta_{\pm})\in\mathcal{I}(f,\theta)$ and that the expression $f=\sum f_{\theta,\mathfrak{a}}$ is reduced. If $\deg(\mathfrak{a})<1$ , we obtain $\mathfrak{a}(f,\theta_{\pm})\prec_{\theta}\mathfrak{a}$ , which contradicts that $\mathfrak{a},\mathfrak{a}(f,\theta_{\pm})\in\mathcal{I}(f,\theta)$ and that the expression $f=\sum f_{\theta,\mathfrak{a}}$ is reduced. Hence, we obtain $\deg(\mathfrak{a})=1$ . Because $\mathfrak{a},\mathfrak{a}(f,\theta_{\pm})\in\mathcal{I}(f,\theta)$ , we obtain $\mathop{\rm Re}\nolimits(\mathfrak{a}_{1}e^{\sqrt{-1}\theta})=\mathop{\rm Re}\nolimits(\sqrt{-1}e^{\sqrt{-1}\theta})$ . If $\mathfrak{a}_{1}\neq\sqrt{-1}$ , then either one of $\mathop{\rm Re}\nolimits(\mathfrak{a}_{1}e^{\sqrt{-1}\theta_{+}})>\mathop{\rm Re}\nolimits(\sqrt{-1}e^{\sqrt{-1}\theta+})$ or $\mathop{\rm Re}\nolimits(\mathfrak{a}_{1}e^{\sqrt{-1}\theta_{-}})>\mathop{\rm Re}\nolimits(\sqrt{-1}e^{\sqrt{-1}\theta-})$ holds. It contradicts $\mathfrak{a}\prec_{\theta_{\pm}}\mathfrak{a}(f,\theta_{\pm})$ . Therefore, we obtain the first claim $\deg(\mathfrak{a}-\sqrt{-1}w)<1$ .

Let us study the second claim. We have the expansion of $\mathfrak{b}\in\mathcal{I}(f,0)$ :

\mathfrak{b}=\sum_{0<\kappa}\mathfrak{b}_{\kappa}w^{\kappa}.

Choose a sufficiently small $\theta_{+}>0$ . Note that

\mathfrak{a}(f,\theta_{+})=\sqrt{-1}w+\sum_{0<\kappa<1}\mathfrak{a}(f,\theta_{+})_{\kappa}w^{\kappa}\in\mathcal{I}(f,0).

If $\deg(\mathfrak{b})<1$ , we obtain $\mathfrak{a}(f,\theta_{+})\prec_{\theta_{+}}\mathfrak{b}$ , which contradicts the choice of $\mathfrak{a}(f,\theta_{+})$ . Hence, we obtain $\deg(\mathfrak{b})\geq 1$ . Because $\mathfrak{b},\mathfrak{a}(f,\theta_{+})\in\mathcal{I}(f,0)$ , we obtain $\mathop{\rm Re}\nolimits(\mathfrak{b}_{\deg(\mathfrak{b})})=0$ . Note that $\mathfrak{b}\prec_{\theta_{+}}\mathfrak{a}(f,\theta_{+})$ . Hence, if $\deg(\mathfrak{b})>1$ , we obtain $\mathop{\rm Re}\nolimits(\mathfrak{b}_{\deg\mathfrak{b}}e^{\sqrt{-1}\deg(\mathfrak{b})\theta_{+}})<0$ . If $\deg(\mathfrak{b})=1$ , we obtain $\mathop{\rm Re}\nolimits\bigl{(}(\mathfrak{b}_{\deg\mathfrak{b}}-\sqrt{-1})e^{\sqrt{-1}\deg(\mathfrak{b})\theta_{+}}\bigr{)}\leq 0$ . Therefore, $\mathfrak{b}_{\deg(\mathfrak{b})}$ satisfies the desired condition. We can check the third claim in a similar way. ∎

For each $\mathfrak{a}=\sum\mathfrak{a}_{k}w^{\kappa}\in\mathcal{I}(f,\theta)$ , we set $\mathfrak{s}(\mathfrak{a}):=\{\kappa\in{\mathbb{R}}_{>0}\,|\,\mathfrak{a}_{\kappa}\neq 0\}$ , and we set

S(f)=\bigcup_{-\delta_{0}<\theta<\pi+\delta_{0}}\bigcup_{\mathfrak{a}\in\mathcal{I}(f,\theta)}\mathfrak{s}(\mathfrak{a})\subset{\mathbb{R}}.

Note that $S(f)$ is a finite subset of ${\mathbb{R}}$ . We choose $0<\epsilon_{0}<1/10$ satisfying the following condition:

\epsilon_{0}<\min\bigl{\{}|\kappa_{1}-\kappa_{2}|/10\,\big{|}\,\kappa_{1},\kappa_{2}\in S,\,\,\kappa_{1}\neq\kappa_{2}\bigr{\}}.

(93)

5.5.2. Another coordinate $g$ and some estimates

Let $A$ be a complex number such that $\mathop{\rm Re}\nolimits(\sqrt{-1}A)>0$ and $\mathop{\rm Re}\nolimits(\sqrt{-1}Ae^{\sqrt{-1}(1-\epsilon_{0})\pi})>0$ . We put $g=w+Aw^{1-\epsilon_{0}}$ on $\{|w|>R_{0},\,\,|\arg(w)-\pi/2|<\pi\}$ . The following lemma is easy to see.

Lemma 5.12.

For any $0<L_{2}<L_{1}<L_{0}<\pi$ , there exist $R_{0}<R_{1}<R_{2}$ and an open subset

\{|w|>R_{2},|\arg(w)-\pi/2|<L_{2}\}\subset\mathcal{U}\subset\{|w|>R_{0},|\arg(w)-\pi/2|<L_{0}\},

such that $g$ induces an isomorphism $\mathcal{U}\simeq\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,|\zeta|>R_{1},\,\,|\arg(\zeta)-\pi/2|<L_{1}\bigr{\}}$ .

Proof We set $\mathcal{U}_{0}:=\{|w|>R_{0},\,|\arg(w)-\pi/2|<L_{0}\}$ . Let $\overline{\mathcal{U}}_{0}$ denote the closure of $\mathcal{U}_{0}$ in $\widetilde{\mathbb{P}}^{1}_{\infty}$ . Note that $g$ induces a continuous map $\overline{\mathcal{U}}_{0}\longrightarrow\widetilde{\mathbb{P}}^{1}_{\infty}$ , which is also denoted by $g$ . For any $Q\in\varpi_{\infty}^{-1}(\infty)\cap\overline{\mathcal{U}}_{0}$ , we obtain $g(Q)=Q$ . Moreover, for the real coordinate systems $(|w|^{-1},\arg(w))$ , the map $g$ is $C^{1}$ around $Q$ , and the derivative at $Q$ is the identity. Then, the claim of the lemma follows from the inverse function theorem. ∎

Lemma 5.13.

There exist $C_{i}>0$ $(i=1,2)$ such that

|f|\leq C_{1}e^{-\mathop{\rm Im}\nolimits(g)}

on $\{0\leq\mathop{\rm Im}\nolimits(g),\,\,|g|>C_{2}\}$ .

Proof It follows from Lemma 5.18 and Lemma 5.19 below. ∎

Lemma 5.14.

There exists $B_{0}>0$ such that $g$ induces a holomorphic isomorphism

\bigl{\{}w\in X(R_{0},\delta_{0})\,\bigr{|}\,\mathop{\rm Im}\nolimits(g(w))\geq B_{0}\bigr{\}}\simeq\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,\mathop{\rm Im}\nolimits(\zeta)\geq B_{0}\bigr{\}}.

Proof Take $\delta_{0}<\delta_{1}<\delta_{2}<\pi/2$ . There exist $R_{0}<R_{1}<R_{2}$ and an open subset $X(R_{2},\delta_{0})\subset\mathcal{U}\subset X(R_{0},\delta_{2})$ such that $g$ induces an isomorphism $\mathcal{U}\simeq X(R_{1},\delta_{1})$ . In particular, $g$ induces an injection on $X(R_{2},\delta_{0})$ . There exists $R_{3}>R_{2}$ such that

g^{-1}\Bigl{(}g\bigl{(}X(R_{0},\delta_{0})\setminus X(R_{2},\delta_{0})\bigr{)}\Bigr{)}\cap X(R_{3},\delta_{0})=\emptyset.

We obtain $g^{-1}\bigl{(}g(X(R_{3},\delta_{0}))\bigr{)}\cap X(R_{0},\delta_{0})=X(R_{3},\delta_{0})$ . By Lemma 5.12, there exists $R_{4}>0$ such that $X(R_{4},0)\subset g(X(R_{3},\delta_{0}))$ . Then, any $B_{0}>R_{4}$ satisfies the desired condition. ∎

5.5.3. Some sectors

For any $R>0$ , $B\in{\mathbb{R}}$ and $0<L<\pi/4$ , we set

S_{0}(g,R,B,L):=\\ \bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,|g(w)|>R,\,\mathop{\rm Im}\nolimits(g(w))>B,\,\arg(g(w))<L\bigr{\}},

(94)

S_{\pi}(g,R,B,L):=\\ \bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,|g(w)|>R,\,\mathop{\rm Im}\nolimits(g(w))>B,\,\arg(g(w))>\pi-L\bigr{\}}.

(95)

Let $B_{0}$ be a positive constant as in Lemma 5.14. By Lemma 5.13, there exists $C>0$ such that

|f|\leq Ce^{-\mathop{\rm Im}\nolimits g}

(96)

on $\{w\in X(R_{0},\delta_{0})\,|\,\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ . Let $R^{(0)}>0$ , $B^{(0)}>B_{0}$ and $0<L^{(0)}<\pi/4$ such that $R^{(0)}\sin(L^{(0)})>10B^{(0)}$ and

S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\subset W(f,0),\quad S_{\pi}(g,R^{(0)},B^{(0)},L^{(0)})\subset W(f,\pi),

(97)

\mathcal{Z}(f)\cap\{0<\theta<2L^{(0)}\}=\emptyset,\quad\mathcal{Z}(f)\cap\{\pi-2L^{(0)}<\theta<\pi\}=\emptyset.

(98)

Lemma 5.15.

For any $B^{(0)}_{1}>B^{(0)}$ and $0<L^{(0)}_{1}<L^{(0)}$ , there exist $R^{(0)}_{1}>R^{(0)}$ and a subset $\mathcal{C}_{0}\subset S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ such that the following holds.

•

There exist $C^{(0)}_{i}>0$ $(i=1,2)$ such that

|f|\geq C^{(0)}_{1}\exp\Bigl{(}\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(f,\theta_{0})\bigr{)}-C^{(0)}_{2}|g|^{\epsilon_{0}/2}\Bigr{)}

on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\setminus\mathcal{C}_{0}$ , where $\theta_{0}>0$ is sufficiently small.

•

Let $\mathcal{D}$ be any connected component of $\mathcal{C}_{0}$ such that

$\mathcal{D}\cap S_{0}(g,R^{(0)}_{1},B^{(0)}_{1},L^{(0)}_{1})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ .
•

There exists an increasing sequence of positive numbers $T_{0,i}$ such that (i) $\lim T_{0,i}=\infty$ , (ii) $\mathcal{C}_{0}\cap\{|g(w)|=T_{0,i}\}=\emptyset$ .

Proof Note that $\mathfrak{a}(f,\theta_{0})$ is contained in $\mathcal{I}(f,0)$ . Consider

\mathfrak{b}\in\mathcal{I}(f,0)\setminus\{\mathfrak{a}(f,\theta_{0})\}.

According to Lemma 5.11, there exist the following three cases; (a) $\deg(\mathfrak{b})>1$ and $-\sqrt{-1}\mathfrak{b}_{\deg\mathfrak{b}}>0$ , (b) $\deg(\mathfrak{b})=1$ and $-\sqrt{-1}\mathfrak{b}_{1}>1$ , (c) $\deg(\mathfrak{b})=1$ and $-\sqrt{-1}\mathfrak{b}_{1}=1$ .

According to the first claim of Lemma 5.19 below, if either (a) or (b) is satisfied, there exist $0<L^{(0)}_{2}(\mathfrak{b})\leq L^{(0)}$ , $C_{1}(\mathfrak{b})>0$ and $C^{\prime}_{1}(\mathfrak{b})>0$ such that $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{1}(\mathfrak{b})|g|^{\epsilon_{0}}+C^{\prime}_{1}(\mathfrak{b})$ holds on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)}_{2}(\mathfrak{b}))$ . Because (a) $\mathop{\rm Re}\nolimits(\mathfrak{b}_{\deg(\mathfrak{b})}e^{\sqrt{-1}\deg(\mathfrak{b})\theta})<0$ or (b) $\mathop{\rm Re}\nolimits((\mathfrak{b}_{\deg(\mathfrak{b})}-\sqrt{-1})e^{\sqrt{-1}\deg(\mathfrak{b})\theta})<0$ holds for any $0<\theta\leq L^{(0)}$ , there exist $0<C_{2}(\mathfrak{b})<C_{1}(\mathfrak{b})$ and $C^{\prime}_{2}(\mathfrak{b})>C^{\prime}_{1}(\mathfrak{b})$ such that $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{2}(\mathfrak{b})|g|^{\epsilon_{0}}+C^{\prime}_{2}(\mathfrak{b})$ holds on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\cap\{\arg(g(w))>L^{(0)}_{2}(\mathfrak{b})/2\}$ . Therefore, we obtain $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{2}(\mathfrak{b})|g|^{\epsilon_{0}}+C_{2}^{\prime}(\mathfrak{b})$ holds on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ .

In the case (c), by the third claim of Lemma 5.19, there exist $0<L^{(0)}_{2}(\mathfrak{b})<L^{(0)}$ , $C_{1}(\mathfrak{b})>0$ and $C_{1}^{\prime}(\mathfrak{b})>0$ such that either $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{1}(\mathfrak{b})|g|^{\epsilon_{0}}+C_{1}^{\prime}(\mathfrak{b})$ or $\mathop{\rm Re}\nolimits(\mathfrak{a}(f,\theta_{0})-\mathfrak{b})\leq-C_{1}(\mathfrak{b})|g|^{\epsilon_{0}}+C_{1}^{\prime}(\mathfrak{b})$ holds on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)}_{2}(\mathfrak{b}))$ . Note that $\mathfrak{b}\prec_{\theta}\mathfrak{a}(f,\theta_{0})$ for $0<\theta\leq L^{(0)}$ by Lemma 4.20 and (98). Hence, $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{1}(\mathfrak{b})|g|^{\epsilon_{0}}+C_{1}^{\prime}(\mathfrak{b})$ holds, indeed. Moreover, because $\mathfrak{b}\prec_{\theta}\mathfrak{a}(f,\theta_{0})$ for $0<\theta\leq L^{(0)}$ , and because $\deg(\mathfrak{a}(f,\theta)-\mathfrak{b})>2\epsilon_{0}$ , there exist $0<C_{2}(\mathfrak{b})<C_{1}(\mathfrak{b})$ and $C_{2}^{\prime}(\mathfrak{b})>C_{1}^{\prime}(\mathfrak{b})$ such that $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{2}(\mathfrak{b})|g|^{\epsilon_{0}}+C_{2}^{\prime}(\mathfrak{b})$ holds on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\cap\{\arg(w)>L^{(0)}_{2}(\mathfrak{b})/2\}$ . Hence, we obtain $\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{2}(\mathfrak{b})|g|^{\epsilon_{0}}+C_{2}^{\prime}(\mathfrak{b})$ on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ .

In all, there exist $C_{3}>0$ and $C_{3}^{\prime}>0$ such that

\mathop{\rm Re}\nolimits(\mathfrak{b}-\mathfrak{a}(f,\theta_{0}))\leq-C_{3}|g|^{\epsilon_{0}}+C_{3}^{\prime}

on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ for any $\mathfrak{b}\in\mathcal{I}(f,0)\setminus\{\mathfrak{a}(f,\theta_{0})\}$ . Hence, there exists $C_{i}>0$ $(i=4,5)$ such that

\Bigl{|}\sum_{\mathfrak{b}\neq\mathfrak{a}(f,\theta_{0})}f_{\mathfrak{b}}\Bigr{|}\leq C_{4}\exp\Bigl{(}\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(f,\theta_{0})\bigr{)}-C_{5}|g|^{\epsilon_{0}}\Bigr{)}.

Then, we obtain the claim of the lemma by applying Lemma 4.38 with $\rho=\epsilon_{0}/2$ . ∎

The following lemma is similar to Lemma 5.15.

Lemma 5.16.

For any $B^{(0)}_{1}>B^{(0)}$ , and $0<L^{(0)}_{1}<L^{(0)}$ , there exist $R^{(0)}_{1}>R^{(0)}$ , and a subset $\mathcal{C}_{\pi}\subset S_{\pi}(g,R^{(0)},B^{(0)},L^{(0)})$ such that the following holds.

•

There exists $C^{(0)}_{i}>0$ $(i=1,2)$ such that

|f|\geq C^{(0)}_{1}\exp\Bigl{(}\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(f,\theta_{\pi})\bigr{)}-C^{(0)}_{2}|g|^{\epsilon_{0}/2}\Bigr{)}

on $S_{\pi}(g,R^{(0)},B^{(0)},L^{(0)})\setminus\mathcal{C}_{\pi}$ , where $\pi-\theta_{\pi}>0$ is sufficiently small.

•

Let $\mathcal{D}$ be any connected component of $\mathcal{C}_{\pi}$ such that

$\mathcal{D}\cap S_{\pi}(g,R^{(0)}_{1},B^{(0)}_{1},L^{(0)}_{1})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $S_{\pi}(g,R^{(0)},B^{(0)},L^{(0)})$ .
•

There exists an increasing sequence of positive numbers $T_{\pi,i}$ such that (i) $\lim T_{\pi,i}=\infty$ , (ii) $\mathcal{C}_{\pi}\cap\{|g(w)|=T_{\pi,i}\}=\emptyset$ . ∎

5.5.4. Relatively compact open subsets

We introduce a condition for a relatively compact open subset $Y$ in $X(R_{0},\delta_{0})$ , which we shall use in §5.6.4–§5.6.7.

Let $B_{0}$ be a constant as in Lemma 5.14. We take $R^{(0)}>0$ , $B^{(0)}>B_{0}$ and $0<L^{(0)}<\pi/4$ as in §5.5.3. We set $B_{1}^{(0)}=2B^{(0)}$ and $L^{(0)}_{1}=\frac{1}{2}L^{(0)}$ , and let $R_{1}^{(0)}>R^{(0)}$ be as in Lemma 5.15 and Lemma 5.16. We take $B_{0}<B^{(0)}_{-1}<B^{(0)}$ , $0<R^{(0)}_{-1}<R^{(0)}$ , and $L^{(0)}<L^{(0)}_{-1}<\pi/4$ such that

S_{0}(g,R^{(0)}_{-1},B^{(0)}_{-1},L^{(0)}_{-1})\subset W(f,0),\quad S_{\pi}(g,R^{(0)}_{-1},B^{(0)}_{-1},L^{(0)}_{-1})\subset W(f,\pi).

We also assume $R^{(0)}_{-1}\sin(L^{(0)})>10B^{(0)}$ .

Condition 5.17.

•

The boundary $\partial Y$ is smooth.

•

There exists $T_{0,i(0)}>2R^{(0)}_{-1}$ such that

Y\cap S_{0}(g,R_{-1}^{(0)},B_{-1}^{(0)},L_{-1}^{(0)})=S_{0}(g,R_{-1}^{(0)},B_{-1}^{(0)},L_{-1}^{(0)})\cap\{|g(w)|\leq T_{0,i(0)}\}.

•

There exists $T_{\pi,i(\pi)}>2R^{(0)}_{-1}$ such that

Y\cap S_{\pi}(g,R_{-1}^{(0)},B_{-1}^{(0)},L_{-1}^{(0)})=S_{\pi}(g,R_{-1}^{(0)},B_{-1}^{(0)},L_{-1}^{(0)})\cap\{|g(w)|\leq T_{\pi,i(\pi)}\}.

•

$\bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,\mathop{\rm Im}\nolimits g(w)\geq B_{-1}^{(0)},\,|g(w)|\leq 2R_{-1}^{(0)}\bigr{\}}\subset Y$ .

We shall use an exhaustive family satisfying Condition 5.17 to construct harmonic metrics with the desired property in Theorem 5.3 and Theorem 6.1.

5.5.5. Appendix: A choice of complex coordinate

Let $R>0$ and $0<\delta<\pi/2$ . Let $\mathfrak{a}_{i}$ $(i=1,\ldots,n(1))$ be mutually distinct holomorphic functions on $X(R,\delta)$ expressed as a finite sum

\mathfrak{a}_{i}=\sqrt{-1}w+\sum_{0<\kappa<1}\mathfrak{a}_{i,\kappa}w^{\kappa}.

Let $\mathfrak{b}_{i}$ $(i=1,\ldots,n(2))$ be holomorphic functions on $X(R,\delta)$ expressed as a finite sum

\mathfrak{b}_{i}=\sum_{0<\kappa\leq\rho(\mathfrak{b}_{i})}\mathfrak{b}_{i,\kappa}w^{\kappa},

where one of the following holds.

•

$\rho(\mathfrak{b}_{i})>1$ and $-\sqrt{-1}\mathfrak{b}_{i,\rho(\mathfrak{b}_{i})}>0$ .
•

$\rho(\mathfrak{b}_{i})=1$ and $-\sqrt{-1}\mathfrak{b}_{i,1}>1$ .

Let $\mathfrak{c}_{i}$ $(i=1,\ldots,n(3))$ be holomorphic functions on $X(R,\delta)$ expressed as a finite sum

\mathfrak{c}_{i}=\sum_{0<\kappa\leq\rho(\mathfrak{c}_{i})}\mathfrak{c}_{i,\kappa}w^{\kappa},

where one of the following holds.

•

$\rho(\mathfrak{c}_{i})>1$ and $-\sqrt{-1}\mathfrak{c}_{i,\rho(\mathfrak{c}_{i})}e^{\pi\sqrt{-1}\rho(\mathfrak{c}_{i})}<0$ .
•

$\rho(\mathfrak{c}_{i})=1$ and $\sqrt{-1}\mathfrak{c}_{i,1}<-1$ .

We set

S=\{0,1\}\cup\bigcup_{i}\{\kappa\,|\,\mathfrak{a}_{i,\kappa}\neq 0\}\cup\bigcup_{i}\{\kappa\,|\,\mathfrak{b}_{i,\kappa}\neq 0\}\cup\bigcup_{i}\{\kappa\,|\,\mathfrak{c}_{i,\kappa}\neq 0\}.

Take $\epsilon>0$ such that $2\epsilon<|\kappa_{1}-\kappa_{2}|$ for any two distinct elements $\kappa_{1},\kappa_{2}\in S$ .

Let $A$ be a complex number such that

\mathop{\rm Re}\nolimits(\sqrt{-1}A)>0,\quad\mathop{\rm Re}\nolimits(\sqrt{-1}Ae^{\sqrt{-1}(1-\epsilon)\pi})>0.

We set $g(w)=w+Aw^{1-\epsilon}$ on $X(R,\delta)$ . In the proof of Lemma 5.14, we observed that there exists $R_{0}>R$ such that (i) $g^{-1}\bigl{(}g(X(R_{0},\delta))\bigr{)}=X(R_{0},\delta)$ , (ii) $g$ induces an isomorphism $X(R_{0},\delta)\simeq g(X(R_{0},\delta))$ . According to Lemma 5.12, there exists $R_{0}^{\prime}>R_{0}$ such that $X(R_{0}^{\prime},\delta/2)\subset g(X(R_{0},\delta))$ .

Lemma 5.18.

There exist positive constants $R_{j}>R^{\prime}_{0}$ $(j=1,2)$ and $C_{1}$ such that the following conditions are satisfied:

•

$\mathop{\rm Re}\nolimits(\mathfrak{a}_{p})-\mathop{\rm Re}\nolimits(\sqrt{-1}g)<-C_{1}|w|^{1-\epsilon}$ for any $p$ on $X(R_{1},0)=\{|w|\geq R_{1},\,\,\,\mathop{\rm Im}\nolimits(w)\geq 0\}$ .
•

$\bigl{\{}w\in X(R,\delta)\,\big{|}\,|g(w)|>R_{2},\,\,\mathop{\rm Im}\nolimits g(w)\geq 0\bigr{\}}\subset X(R_{1},0)$ .

Proof Note that $\sqrt{-1}g-\mathfrak{a}_{p}=\sqrt{-1}Aw^{1-\epsilon}+\sum_{0<\kappa<1-\epsilon}\alpha_{p,\kappa}w^{\kappa}$ for $\alpha_{p,\kappa}\in{\mathbb{C}}$ . Because $\{(p,\kappa)\,|\,\alpha_{p,\kappa}\neq 0\}$ is finite, there exist $R_{1}$ and $C_{1}$ as in the first condition.

By our choice of $R_{0}<R_{0}^{\prime}$ , the set

\mathcal{U}_{1}:=\bigl{\{}w\in X(R,\delta)\,\big{|}\,|g(w)|>R^{\prime}_{0},\,\,\mathop{\rm Im}\nolimits g(w)\geq 0\bigr{\}}\simeq\\ \bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,|\zeta|>R^{\prime}_{0},\,\mathop{\rm Im}\nolimits\zeta\geq 0\bigr{\}}

(99)

is connected. Clearly, we obtain $\mathcal{U}_{1}\cap X(R,0)\neq\emptyset$ . Because $\mathop{\rm Im}\nolimits g(w)=\mathop{\rm Im}\nolimits(w)-\mathop{\rm Re}\nolimits(\sqrt{-1}Aw^{1-\epsilon})$ , the intersection

\bigl{\{}w\in X(R,\delta)\,\big{|}\,|g(w)|>R^{\prime}_{0},\mathop{\rm Im}\nolimits g(w)\geq 0\bigr{\}}\cap\{w\in X(R,\delta)\,|\,\mathop{\rm Im}\nolimits(w)=0\}

is empty. Hence, if $R_{2}>R_{0}^{\prime}$ is sufficiently large, we obtain $\bigl{\{}w\in X(R,\delta)\,\big{|}\,|g(w)|>R_{2},\,\,\mathop{\rm Im}\nolimits g(w)\geq 0\bigr{\}}\subset X(R_{1},0)$ . ∎

Lemma 5.19.

There exist positive constants $L$ , $R_{3}$ and $C_{j}$ $(j=2,3)$ such that the following holds:

•

$\mathop{\rm Re}\nolimits(\mathfrak{b}_{i})-\mathop{\rm Re}\nolimits(\mathfrak{a}_{p})<-C_{2}|g|^{\rho(\mathfrak{b}_{i})-\epsilon}$ for any $i$ and $p$ on

\mathcal{U}_{0}:=\bigl{\{}w\in X(R,\delta)\,\big{|}\,\mathop{\rm Re}\nolimits g(w)>0,\,\,|g(w)|\geq R_{3},\,\,0\leq\arg g(w)\leq L\bigr{\}}.

•

$\mathop{\rm Re}\nolimits(\mathfrak{c}_{i})-\mathop{\rm Re}\nolimits(\mathfrak{a}_{p})<-C_{2}|g|^{\rho(\mathfrak{c}_{i})-\epsilon}$ for any $i$ and $p$ on

\mathcal{U}_{\pi}:=\bigl{\{}w\in X(R,\delta)\,\big{|}\,\mathop{\rm Re}\nolimits g(w)<0,\,\,|g(w)|\geq R_{3},\,\,\pi-L\leq\arg g(w)\leq\pi\bigr{\}}.

•

For $p\neq q$ , we obtain $C_{3}|g|^{\epsilon}\leq\bigl{|}\mathop{\rm Re}\nolimits(\mathfrak{a}_{p}-\mathfrak{a}_{q})\bigr{|}$ on $\mathcal{U}_{0}\cup\mathcal{U}_{\pi}$ .

Proof We use the following estimate on $X(R_{0},\delta)$ for some $\epsilon^{\prime}>0$ :

w-\bigl{(}g-Ag^{1-\epsilon}\bigr{)}=O(|w|^{1-\epsilon-\epsilon^{\prime}}).

(100)

Let us study the first claim. To simplify the notation, we set $\rho(i):=\rho(\mathfrak{b}_{i})$ . If $\rho(i)>1$ , we obtain the following expression

\mathfrak{b}_{i}-\mathfrak{a}_{p}=\mathfrak{b}_{i,\rho(i)}w^{\rho(i)}+\sum_{0<\kappa<\rho(i)}\mathfrak{b}_{(i,p),\kappa}w^{\kappa}.

By (100), there exists $\delta(i,p)>0$ such that

\mathfrak{b}_{i}-\mathfrak{a}_{p}-\Bigl{(}\mathfrak{b}_{i,\rho(i)}g^{\rho(i)}-\rho(i)\cdot\mathfrak{b}_{i,\rho(i)}Ag^{\rho(i)-\epsilon}\Bigr{)}=O(|g|^{\rho(i)-\epsilon-\delta(i,p)}).

By the assumption, there exists $b>0$ such that $\mathfrak{b}_{i,\rho(i)}=\sqrt{-1}b$ . We obtain

\mathop{\rm Re}\nolimits\Bigl{(}\mathfrak{b}_{i,\rho(i)}g^{\rho(i)}-\rho(i)\cdot\mathfrak{b}_{i,\rho(i)}Ag^{\rho(i)-\epsilon}\Bigr{)}=\\ -b|g|^{\rho(i)}\sin\bigl{(}\rho(i)\arg(g)\bigr{)}-b\rho(i)|g|^{\rho(i)-\epsilon}\mathop{\rm Re}\nolimits\bigl{(}\sqrt{-1}Ae^{\sqrt{-1}(\rho(i)-\epsilon)\arg(g)}\bigr{)}.

(101)

There exists $L_{1}(i,p)>0$ such that

\sin\bigl{(}\rho(i)\arg(g)\bigr{)}\geq 0,\quad\mathop{\rm Re}\nolimits\bigl{(}\sqrt{-1}Ae^{\sqrt{-1}(\rho(i)-\epsilon)\arg(g)}\bigr{)}>0

on $\bigl{\{}0\leq\arg(g)\leq L_{1}(i,p)\bigr{\}}$ . Hence, there exist $R_{2}^{\prime}>0$ and $C_{3}(i,p)>0$ such that

\mathop{\rm Re}\nolimits(\mathfrak{b}_{i}-\mathfrak{a}_{p})\leq-C_{3}(i,p)|g|^{\rho(i)-\epsilon}

(102)

on $\bigl{\{}w\in X(R,\delta)\,\big{|}\,|g(w)|>R_{2}^{\prime},\,\,0\leq\arg(g)\leq L_{1}(i,p)\bigr{\}}$ . If $\rho(i)=1$ , there exists $b>1$ such that $\mathfrak{b}_{i,1}=\sqrt{-1}b$ . We obtain

\mathfrak{b}_{i}-\mathfrak{a}_{p}=(\mathfrak{b}_{i,1}-\sqrt{-1})w+\sum_{0<\kappa<1}\mathfrak{b}_{(i,p),\kappa}w^{\kappa}\\ =\sqrt{-1}(b-1)w+\sum_{0<\kappa<1}\mathfrak{b}_{(i,p),\kappa}w^{\kappa}.

(103)

By the same argument, we can prove that there exist $L_{1}(i,p)>0$ , $R_{2}^{\prime}>0$ and $C_{3}(i,p)>0$ such that (102) holds on $\bigl{\{}w\in X(R,\delta)\,\big{|}\,|g(w)|>R_{2}^{\prime},\,\,0\leq\arg(g)\leq L_{1}(i,p)\bigr{\}}$ with $\rho(i)=1$ . Thus, we obtain the first claim. The second claim can be proved similarly.

Let us prove the third claim. For $p\neq q$ , we set $\rho(p,q):=\deg(\mathfrak{a}_{p}-\mathfrak{a}_{q})>0$ . We obtain the following expression:

\mathfrak{a}_{p}-\mathfrak{a}_{q}=\sum_{0<\kappa\leq\rho(p,q)}\mathfrak{a}_{(p,q),\kappa}w^{\kappa}.

Here, $\mathfrak{a}_{(p,q),\rho(p,q)}\neq 0$ . There exists $\delta(p,q)>0$ such that the following holds:

\mathop{\rm Re}\nolimits(\mathfrak{a}_{p}-\mathfrak{a}_{q})-\mathop{\rm Re}\nolimits\Bigl{(}\mathfrak{a}_{(p,q),\rho(p,q)}\bigl{(}g^{\rho(p,q)}-\rho(p,q)Ag^{\rho(p,q)-\epsilon}\bigr{)}\Bigr{)}\\ =O\bigl{(}|g|^{\rho(p,q)-\epsilon-\delta(p,q)}\bigr{)}.

(104)

If $\mathfrak{a}_{p,q,\rho(p,q)}\not\in\sqrt{-1}{\mathbb{R}}$ , there exist $L_{3}(p,q)>0$ and $C_{4}(p,q)>0$ such that

\Bigl{|}\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}_{(p,q),\rho(p,q)}g^{\rho(p,q)}\bigr{)}\Bigr{|}>C_{4}(p,q)|g|^{\rho(p,q)}

(105)

on $\bigl{\{}w\in X(R_{0},\delta)\,\big{|}\,0\leq\arg(g(w))\leq L_{3}(p,q)\bigr{\}}$ . If $\mathfrak{a}_{p,q,\rho(p,q)}\in\sqrt{-1}{\mathbb{R}}\setminus\{0\}$ , there exists $b\neq 0$ such that $\mathfrak{a}_{p,q,\rho(p,q)}=\sqrt{-1}b$ . Note that

\Bigl{|}\mathop{\rm Re}\nolimits\Bigl{(}\mathfrak{a}_{(p,q),\rho(p,q)}\bigl{(}g^{\rho(p,q)}-\rho(p,q)Ag^{\rho(p,q)-\epsilon}\bigr{)}\Bigr{)}\Bigr{|}=|b|\times\\ \Bigl{|}|g|^{\rho(p,q)}\sin\bigl{(}\rho(p,q)\arg g\bigr{)}+\rho(p,q)|g|^{\rho(p,q)-\epsilon}\mathop{\rm Re}\nolimits\bigl{(}\sqrt{-1}Ae^{\sqrt{-1}(\rho(p,q)-\epsilon)\arg(g)}\bigr{)}\Bigr{|}.

(106)

Hence, there exist $L_{3}(p,q)>0$ and $C_{4}(p,q)>0$

\Bigl{|}\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}_{(p,q),\rho(p,q)}g^{\rho(p,q)}\bigr{)}\Bigr{|}>C_{4}(p,q)|g|^{\rho(p,q)-\epsilon}.

on $\bigl{\{}w\in X(R_{0},\delta)\,\big{|}\,0\leq\arg(g(w))\leq L_{3}(p,q)\bigr{\}}$ . Then, we can easily deduce the third claim. ∎

5.6. Proof of Theorem 5.3 and Proposition 5.4

Take $R_{0}>10$ and $0<\delta_{0}<\pi/2$ . Let $f$ be a holomorphic function on $X(R_{0},\delta_{0})$ as in §5.5.1. We set $q=f\,(dw)^{r}$ . We first study $G_{r}$ -invariant harmonic metrics of $(\mathbb{K}_{X(R_{0},\delta_{0})},\theta(q))$ in §5.6.1–§5.6.6. Then, we shall derive Theorem 5.3 and Proposition 5.4 in §5.6.7.

5.6.1. The associated parabolic weights

Let $\epsilon_{0}>0$ be as in §5.5.1.

Proposition 5.20.

Let $h\in\mathop{\rm Harm}\nolimits(q)$ . Then, there exists ${\boldsymbol{a}}(h)\in\mathcal{P}$ satisfying the following estimate on any $\bigl{\{}w\in{\mathbb{C}}\,\big{|}\,|w|>R_{1},|\arg(w)-\pi/2|<(1-\delta)\pi/2\bigr{\}}$ for any $R_{1}>R_{0}$ and $\delta>0$ :

\log\bigl{|}(dw)^{(r+1)/2-i}\bigr{|}_{h}=a_{i}(h)\cdot\mathop{\rm Im}\nolimits(w)+O\Bigl{(}|w|^{1-\epsilon_{0}}\Bigr{)}.

(107)

Proof For $R>0$ , $B>0$ and $0<L<\pi/4$ , let $\widetilde{S}(w,R,B,L)$ denote the following set:

\bigl{\{}w\in{\mathbb{C}}\,\big{|}\,|w|>R,\,\mathop{\rm Im}\nolimits(w)\geq B,\,\arg(w)<L\bigr{\}}\cup\\ \bigl{\{}w\in{\mathbb{C}}\,\big{|}\,|w|>R,\,\mathop{\rm Im}\nolimits(w)\geq B,\,\arg(w)>\pi-L\bigr{\}}.

(108)

By using the function $g=w+Aw^{1-\epsilon_{0}}$ as in §5.5.2, from now on we use $w$ to refer to $g$ for simplicity. By the results in §5.5.2, the following condition is satisfied.

Condition 5.21.

There exists $C^{(1)}_{0}>0$ such that $|f|\leq C^{(1)}_{0}e^{-\mathop{\rm Im}\nolimits(w)}$ on $\{|w|>R_{0},\,\,\mathop{\rm Im}\nolimits(w)\geq 0\}$ . Moreover, there exist $2R_{0}<R^{(1)}<R^{(1)}_{1}$ , $2R_{0}<B^{(1)}<B^{(1)}_{1}$ , $0<L^{(1)}_{1}<L^{(1)}$ , and $\mathcal{C}\subset\widetilde{S}(w,R^{(1)},B^{(1)},L^{(1)})$ such that the following conditions are satisfied.

•

There exists $C^{(1)}_{i}>0$ $(i=1,2)$ such that

$|f|\geq C^{(1)}_{1}\exp(-\mathop{\rm Im}\nolimits(w)-C^{(1)}_{2}|w|^{1-\epsilon_{0}})$

on $\widetilde{S}(w,R^{(1)},B^{(1)},L^{(1)})\setminus\mathcal{C}$ .
•

Let $\mathcal{D}$ be any connected component of $\mathcal{C}$ such that

$\mathcal{D}\cap\widetilde{S}(w,R^{(1)}_{1},B^{(1)}_{1},L^{(1)}_{1})\neq\emptyset.$

Then, $\mathcal{D}$ is relatively compact in $\widetilde{S}(w,R^{(1)},B^{(1)},L^{(1)})$ .

Let $\theta(q)$ denote the Higgs field of $\mathbb{K}_{X(R_{0},\delta_{0}),r}$ associated with $q$ . For $w_{1}$ with $\mathop{\rm Im}\nolimits(w_{1})\geq 2R_{0}$ , $\{|w-w_{1}|<\mathop{\rm Im}\nolimits(w_{1})-R_{0}/2\}$ is contained in $\{\mathop{\rm Im}\nolimits(w)>R_{0}\}$ . There exists an isomorphism

\Phi_{w_{1}}:\{|\eta|<1\}\simeq\{|w-w_{1}|<\mathop{\rm Im}\nolimits(w_{1})-R_{0}/2\}

given by $\Phi_{w_{1}}(\eta)=w_{1}+(\mathop{\rm Im}\nolimits(w_{1})-R_{0}/2)\eta$ . Note that there exists $C^{(1)}_{3}>0$ such that $|\Phi_{w_{1}}^{\ast}(q)|<C^{(1)}_{3}$ independently from $w_{1}$ . By applying Proposition 3.7 to $\Phi_{w_{1}}^{\ast}(\mathbb{K}_{X(R_{0},\delta_{0}),r},\theta(q),h)$ , we obtain that the sup norm of $\Phi_{w_{1}}^{\ast}(\theta(q))$ with respect to $\Phi_{w_{1}}^{\ast}(h)$ on $\{|\eta|<1/2\}$ is dominated by a constant independently from $w_{1}$ . Because $\Phi_{w_{1}}^{\ast}(dw)=(\mathop{\rm Im}\nolimits(w_{1})-R_{0}/2)d\eta$ , we obtain $|\theta(q)|_{h}=O\bigl{(}\mathop{\rm Im}\nolimits(w)^{-1}\bigr{)}$ on $\{\mathop{\rm Im}\nolimits(w)\geq 2R_{0}\}$ . By the Hitchin equation, we also obtain $|R(h)|_{h}=O\bigl{(}\mathop{\rm Im}\nolimits(w)^{-2}\bigr{)}$ on $\{\mathop{\rm Im}\nolimits(w)\geq 2R_{0}\}$ .

Lemma 5.22.

The following estimates hold on $\widetilde{S}(w,R^{(1)}_{1},B^{(1)}_{1},L^{(1)}_{1})$ for $i=1,\ldots,r$ :

\log\bigl{|}(dw)^{(r+1)/2-i}\bigr{|}_{h}=O\Bigl{(}\mathop{\rm Im}\nolimits(w)+|w|^{1-\epsilon_{0}}\Bigr{)}.

(109)

Proof We obtain the estimate (109) on $\widetilde{S}(w,R^{(1)},B^{(1)},L^{(1)})\setminus\mathcal{C}$ by the estimate $|\theta(q)|_{h}=O\bigl{(}\mathop{\rm Im}\nolimits(w)^{-1}\bigr{)}$ , Condition 5.21 and Corollary 3.5. Because $|R(h)|=O(\mathop{\rm Im}\nolimits(w)^{-2})$ on $\{\mathop{\rm Im}\nolimits(w)\geq 2R_{0}\}$ , there exist functions $\beta_{i}$ on $\{\mathop{\rm Im}\nolimits(w)\geq 2R_{0}\}$ such that $|\beta_{i}|=O\bigl{(}\log(\mathop{\rm Im}\nolimits(w))\bigr{)}$ and $\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}-\beta_{i}$ are harmonic functions on $\{\mathop{\rm Im}\nolimits(w)\geq 2R_{0}\}$ . (See Lemma 4.5.) Then, by using Condition 5.21 again, we obtain the estimate (109) on $\widetilde{S}(w,R^{(1)}_{1},B^{(1)}_{1},L^{(1)}_{1})$ . ∎

Lemma 5.23.

Let $K$ be any compact subset of $]0,\pi[$ , which we regard as a closed subset in $\varpi_{\infty}^{-1}(\infty)\cap\overline{X}(R_{0},\delta_{0})$ . Then, there exists a neighbourhood $\mathcal{U}$ of $K$ in $\overline{X}(R_{0},\delta_{0})$ such that the estimate (109) holds on $\mathcal{U}\setminus\varpi_{\infty}^{-1}(\infty)$ .

Proof It is enough to study the case where $K$ consists of a point. If $K\subset]0,\pi[\setminus\mathcal{Z}(f)$ , we obtain the claim from Lemma 5.7. If $K\subset\mathcal{Z}(f)\cap]0,\pi[$ , we obtain the claim from Lemma 5.10 and Corollary 4.3. (See the proof of Theorem 5.2 in §5.3.2 for a more detailed argument.) ∎

By Lemma 5.22 and Lemma 5.23, the estimate (109) holds on $\bigl{\{}|w|>R^{(1)}_{1},\mathop{\rm Im}\nolimits(w)\geq B^{(1)}_{1}\bigr{\}}$ . We also have the following estimate on $\{\mathop{\rm Im}\nolimits(w)\geq 2R_{0}\}$ :

\partial\overline{\partial}\log\bigl{|}(dw)^{(r+1)/2-i}\bigr{|}_{h}=O\bigl{(}\mathop{\rm Im}\nolimits(w)^{-2}\bigr{)}.

By Proposition 4.4, there exist $a_{i}(h)\in{\mathbb{R}}$ $(i=1,\ldots,r)$ such that

\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}-a_{i}(h)\mathop{\rm Im}\nolimits(w)=O(|w|^{1-\epsilon_{0}})

on $\{|w|>R^{(1)}_{1},\,\mathop{\rm Im}\nolimits(w)\geq B_{1}^{(1)}\}$ . Because the above $w$ is referring to $g$ , it means

\log\bigl{|}(dg(w))^{(r+1-2i)/2}\bigr{|}_{h}-a_{i}(h)\mathop{\rm Im}\nolimits(g(w))=O(|g(w)|^{1-\epsilon_{0}})

on $\{|g(w)|>R^{(1)}_{1},\,\mathop{\rm Im}\nolimits g(w)\geq B_{1}^{(1)}\}$ . By Lemma 5.12, for any $\delta>0$ , there exists $R_{1}>0$ such that $\{|w|>R_{1},\,\,|\arg(w)-\pi/2|<(1-\delta)\pi/2\}\subset\{|g(w)|>R^{(1)}_{1},\mathop{\rm Im}\nolimits g(w)\geq B_{1}^{(1)}\}$ . Using $dg/dw-1=O(|w|^{-\epsilon_{0}})$ and $\mathop{\rm Im}\nolimits g(w)-\mathop{\rm Im}\nolimits w=O(|w|^{1-\epsilon_{0}})$ , we obtain

\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}-a_{i}(h)\mathop{\rm Im}\nolimits(w)=O(|w|^{1-\epsilon_{0}})

on $\{|w|>R_{1},\,\,|\arg(w)-\pi/2|<(1-\delta)\pi/2\}$ . Because

\theta(q)(dw)^{(r+1-2i)/2}=(dw)^{(r+1-2(i+1))/2}\,dw\quad(i=1,\ldots,r-1)

and $|\theta(q)|_{h}=O\bigl{(}\mathop{\rm Im}\nolimits(w)^{-1}\bigr{)}$ , we obtain $a_{i}(h)\geq a_{i+1}(h)$ . We also obtain $a_{r}(h)\geq a_{1}(h)-1$ by using

\theta(q)(dw)^{(-r+1)/2}=f(dw)^{(r-1)/2}(dw)

and the estimate of $f$ around any point of $(\{\infty\}\times]0,\pi[)\setminus\mathcal{Z}(f)$ as in Condition 5.5. Thus, the proof of Proposition 5.20 is completed. ∎

5.6.2. Mutually boundedness

Proposition 5.24.

Let $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ such that ${\boldsymbol{a}}(h_{1})={\boldsymbol{a}}(h_{2})$ . Then, for any $R_{1}>R_{0}$ and $0<\delta_{1}<\delta_{0}$ , $h_{1}$ and $h_{2}$ are mutually bounded on $X(R_{1},\delta_{1})$ .

Proof Take any $0<\delta_{1}<\delta_{0}$ and $R_{1}>R_{0}$ . It is enough to prove that there exists $R^{\prime}_{1}>R_{1}$ such that $h_{1}$ and $h_{2}$ are mutually bounded on $X(R^{\prime}_{1},\delta_{1})$ because $X(R_{1},\delta_{1})\setminus X(R_{1}^{\prime},\delta_{1})$ is compact.

Let $s$ be determined by $h_{2}=h_{1}\cdot s$ . Because $\det(s)=1$ , we obtain $\mathop{\rm Tr}\nolimits(s)\geq r$ . By Proposition 5.1, $\mathop{\rm Tr}\nolimits(s)$ is bounded on $X(R_{1},\delta_{1})\setminus X(R_{1},\delta_{3})$ for any $0<\delta_{3}<\delta_{1}$ .

By the assumption ${\boldsymbol{a}}(h_{1})={\boldsymbol{a}}(h_{2})$ , the following estimate holds on $\bigl{\{}|w|>R_{1},\,\,|\arg(w)-\pi/2|<(1-\delta)\pi/2\bigr{\}}$ for any $\delta>0$ :

\log\mathop{\rm Tr}\nolimits(s)=O\bigl{(}|w|^{1-\epsilon_{0}}\bigr{)}.

By Lemma 5.10, there exist $N>0$ and a subset $Z_{1}\subset{\mathbb{R}}_{>0}$ with $\int_{Z_{1}}dt/t<\infty$ such that

\log\mathop{\rm Tr}\nolimits(s)=O(|w|^{N})

on $\bigl{\{}|w|>R_{1},\,\,-\delta<\arg(w)<\delta,\,\,|w|\not\in Z_{1}\bigr{\}}$ for any $\delta>0$ . By Corollary 4.3, we obtain that

\log\mathop{\rm Tr}\nolimits(s)=O\bigl{(}|w|^{1-\epsilon_{0}}\bigr{)}

on $\bigl{\{}|w|>R_{1},\,\,-\delta_{1}<\arg(w)<\delta_{1}\bigr{\}}$ . Similarly, we obtain

\log\mathop{\rm Tr}\nolimits(s)=O\bigl{(}|w|^{1-\epsilon_{0}}\bigr{)}

on $\{|w|>R_{1},\,\,\pi-\delta_{1}<\arg(w)<\pi+\delta_{1}\}$ . If $0<\delta_{4}\leq\delta_{1}$ is sufficiently small, we obtain that $\log\mathop{\rm Tr}\nolimits(s)$ is bounded on $X(R_{1},\delta_{4})$ by Corollary 4.2. Because $\log\mathop{\rm Tr}\nolimits(s)$ is bounded on $X(R_{1},\delta_{1})\setminus X(R_{1},\delta_{4}/2)$ , we obtain that $\log\mathop{\rm Tr}\nolimits(s)$ is bounded on $X(R_{1},\delta_{1})$ . ∎

5.6.3. Auxiliary metrics

To introduce an auxiliary metric $h_{0}$ , we set $q_{0}:=e^{\sqrt{-1}g(w)}(dg(w))^{r}$ on $X(R_{0},\delta_{0})$ .

Lemma 5.25.

For any ${\boldsymbol{a}}\in\mathcal{P}$ , there exists $h_{0}\in\mathop{\rm Harm}\nolimits(q_{0})$ such that the following holds on $\mathcal{U}_{1}:=\bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,\mathop{\rm Im}\nolimits(g(w))\geq B_{0}\bigr{\}}$ :

\log\bigl{|}(dw)^{(r+1)/2-i}\bigr{|}_{h_{0}}=a_{i}\cdot\mathop{\rm Im}\nolimits(w)+O\Bigl{(}|w|^{1-\epsilon_{0}}\Bigr{)}.

We also obtain

|\theta(q)|_{h_{0}}=O\Bigl{(}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-1}\Bigr{)},\quad|F(h_{0},\theta(q))|_{h_{0}}=O\Bigl{(}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-2}\Bigr{)}

on $\mathcal{U}_{1}$ . Here, $F(h_{0},\theta(q))=R(h_{0})+[\theta(q),\theta(q)^{\dagger}_{h_{0}}]$ .

Proof We consider the map $\Phi:{\mathbb{C}}\longrightarrow{\mathbb{C}}^{\ast}$ by $\Phi(w)=e^{\sqrt{-1}g(w)/r}$ . We set $q_{1}:=(-\sqrt{-1}rdz)^{r}$ on ${\mathbb{C}}^{\ast}$ . We obtain $q_{0}=\Phi^{\ast}(q_{1})$ . Note that $dg/dw-1=O(|w|^{-\epsilon_{0}})$ and $\mathop{\rm Im}\nolimits(g(w))-y=O(|w|^{1-\epsilon_{0}})$ . Then, the first claim is reduced to [30]. (See Proposition 3.25.) We obtain $|\theta(q_{0})|_{h_{0}}=O\Bigl{(}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-1}\Bigr{)}$ . Because $|f|=O\bigl{(}|e^{\sqrt{-1}g(w)}|\bigr{)}$ , we obtain $|\theta(q)|_{h_{0}}=O\bigl{(}|\theta(q_{0})|_{h_{0}}\bigr{)}$ . By the Hitchin equation $R(h_{0})+[\theta(q_{0}),\theta(q_{0})^{\dagger}_{h_{0}}]=0$ , we obtain $R(h_{0})=O\Bigl{(}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-2}\Bigr{)}$ . Hence, we obtain $|F(h_{0},\theta(q))|_{h_{0}}=O\Bigl{(}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-2}\Bigr{)}$ . Thus, we obtain the second claim. ∎

5.6.4. Comparison with auxiliary metrics

Let $h_{0}$ be as in Lemma 5.25. For any relatively compact open subset $Y\subset X(R_{0},\delta_{0})$ satisfying Condition 5.17, there uniquely exists $h_{Y}\in\mathop{\rm Harm}\nolimits(q_{|Y})$ such that $h_{Y|\partial Y}=h_{0|\partial Y}$ , according to Proposition 2.1.

Lemma 5.26.

There exists $C^{(2)}_{0}>0$ which is independent of $Y$ , such that the following holds on the region $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\cap Y$ :

|\theta(q)|_{h_{Y}}\leq C^{(2)}_{0}\mathop{\rm Im}\nolimits(g(w))^{-1}.

(110)

Proof Because $h_{Y|\partial Y}=h_{0|\partial Y}$ , there exists $C^{(2)}_{1}>0$ , which is independent of $Y$ , such that the following holds on $\partial Y\cap S_{0}(g,R_{-1}^{(0)},B_{-1}^{(0)},L_{-1}^{(0)})$ :

|\theta(q)|_{h_{Y}}\leq C^{(2)}_{1}(\mathop{\rm Im}\nolimits(g(w))^{-1}).

We set

D:=\\ \frac{1}{10}\min\bigl{\{}B^{(0)}-B^{(0)}_{-1},R^{(0)}-R^{(0)}_{-1},R^{(0)}_{-1}\sin(L^{(0)}_{-1}-L^{(0)}),R^{(0)}_{-1}\sin(L^{(0)}_{-1})-B^{(0)}\bigr{\}}.

(111)

Take any $w_{1}\in S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\cap Y$ such that

B^{(0)}\leq\mathop{\rm Im}\nolimits(g(w_{1}))\leq B^{(0)}+D.

We set

\mathcal{B}(w_{1}):=\bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,|g(w)-g(w_{1})|<D\bigr{\}}\simeq\\ \bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,|\zeta-g(w_{1})|<D\bigr{\}}.

(112)

Note that $\mathcal{B}(w_{1})\subset S_{0}(g,R^{(0)}_{-1},B^{(0)}_{-1},L^{(0)}_{-1})$ . By applying Proposition 3.11 to $h_{Y|Y\cap\mathcal{B}(w_{1})}$ , we obtain that there exists $C_{2}^{(2)}>0$ , which is independent of $Y$ and $w_{1}$ as above, such that the following holds on $Y\cap\bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,|g(w)-g(w_{1})|<D/2\bigr{\}}$ :

\bigl{|}\theta(q)\bigr{|}_{h_{Y}}\leq C_{2}^{(2)}.

Take any $w_{1}\in S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ such that $\mathop{\rm Im}\nolimits g(w_{1})>D+B^{(0)}$ and that $\arg g(w_{1})<L^{(0)}_{-1}/2$ . We set $\rho(w_{1}):=(\mathop{\rm Im}\nolimits g(w_{1})-B^{(0)})/2$ . Then, $\mathcal{B}(w_{1}):=\bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,|g(w)-g(w_{1})|<\rho(w_{1})\bigr{\}}$ is contained in the following set:

S_{0}(g,R^{(0)}_{-1},B^{(0)}_{-1},L^{(0)}_{-1})\cup\\ \bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,\mathop{\rm Im}\nolimits g(w)\geq B^{(0)}_{-1},|g(w)|\leq 2R^{(0)}_{-1}\bigr{\}}.

(113)

Let $\Psi_{w_{1}}$ be the isomorphism $\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,|\zeta-g(w_{1})|<\rho(w_{1})\bigr{\}}\simeq\{|\eta|<1\}$ determined by $\Psi_{w_{1}}(\zeta)=\rho(w_{1})^{-1}\bigl{(}\zeta-g(w_{1})\bigr{)}$ . Note that there exists $C^{(2)}_{3}\geq 1$ , which is independent of $Y$ and $w_{1}$ as above, such that

(C^{(2)}_{3})^{-1}\leq\frac{\bigl{|}(\Psi_{w_{1}}\circ g)^{\ast}(d\eta)/dw\bigr{|}}{\mathop{\rm Im}\nolimits g(w_{1})-B^{(0)}}\leq C^{(2)}_{3}.

h_{w_{1}}\in\mathop{\rm Harm}\nolimits\bigl{(}q_{w_{1}|\{|\eta|<1\}\cap(\Psi_{w_{1}}\circ g)(Y)}\bigr{)}

such that $(\Psi_{w_{1}}\circ g)^{\ast}(h_{w_{1}})=h_{Y}$ on $Y\cap\mathcal{B}(w_{1})$ . By applying Proposition 3.11 to $h_{w_{1}|\{|\eta|<1\}\cap(\Psi_{w_{1}}\circ g)(Y)}$ , we obtain that there exists $C^{(2)}_{5}>0$ , which is independent of $Y$ and $w_{1}$ as above, such that the following holds on $\{|\eta|<1/2\}\cap(\Psi_{w_{1}}\circ g)(Y)$ :

\bigl{|}\theta(q_{w_{1}})\bigr{|}_{h_{w_{1}}}\leq C^{(2)}_{5}.

Hence, there exists $C^{(2)}_{6}>0$ , which is independent of $Y$ and $w_{1}$ as above, such that the following holds on $\bigl{\{}|g(w)-g(w_{1})|<\rho(w_{1})/2\bigr{\}}\cap Y$ :

\bigl{|}\theta(q)\bigr{|}_{h_{Y}}\leq C^{(2)}_{6}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-1}.

Take any $w_{1}\in S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ such that $\arg g(w_{1})>L^{(0)}/2$ . We set

\rho(w_{1}):=|g(w_{1})|\sin\bigl{(}(L^{(0)}_{-1}-\arg w_{1})/2\bigr{)}.

Then, $\mathcal{B}(w_{1})=\Bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,|g(w)-g(w_{1})|<\rho(w_{1})\Bigr{\}}$ is contained in (113). Let $\Psi_{w_{1}}$ be the isomorphism $\bigl{\{}\zeta\in{\mathbb{C}}\,\big{|}\,|\zeta-g(w_{1})|<\rho(w_{1})\bigr{\}}\simeq\{|\eta|<1\}$ determined by $\Psi_{w_{1}}(\zeta)=\rho(w_{1})^{-1}(\zeta-g(w_{1}))$ . Note that there exists $C^{(2)}_{7}\geq 1$ , which is independent of $Y$ and $w_{1}$ such that

(C^{(2)}_{7})^{-1}\leq\frac{\bigl{|}(\Psi_{w_{1}}\circ g)^{\ast}(d\eta)/dw\bigr{|}}{|g(w_{1})|}\leq C^{(2)}_{7}.

There exists $q_{w_{1}}$ on $\bigl{\{}|\eta|<1\bigr{\}}$ such that $(\Psi_{w_{1}}\circ g)^{\ast}(q_{w_{1}})=q$ on $\mathcal{B}(w_{1})$ . By (96), there exists $C^{(2)}_{8}>0$ , which is independent of $Y$ and $w_{1}$ , such that $|q_{w_{1}}|\leq C^{(2)}_{8}$ . We also obtain $h_{w_{1}}\in\mathop{\rm Harm}\nolimits\bigl{(}q_{w_{1}|\{|\eta|<1\}\cap(\Psi_{w_{1}}\circ g)(Y)}\bigr{)}$ such that $(\Psi_{w_{1}}\circ g)^{\ast}(h_{w_{1}})=h_{Y}$ on $Y\cap\mathcal{B}(w_{1})$ . By applying Proposition 3.11 to $h_{w_{1}|\{|\eta|<1\}\cap(\Psi_{w_{1}}\circ g)(Y)}$ , we obtain that there exists $C^{(2)}_{9}>0$ , which is independent of $Y$ and $w_{1}$ , such that the following holds on $\{|\eta|<1/2\}\cap(\Psi_{w_{1}}\circ g)(Y)$ :

\bigl{|}\theta(q_{w_{1}})\bigr{|}_{h_{w_{1}}}\leq C^{(2)}_{9}.

Hence, there exists $C^{(2)}_{10}>0$ which is independent of $Y$ and $w_{1}$ such that the following holds on $\bigl{\{}|g(w)-g(w_{1})|<\rho(w_{1})/2\bigr{\}}\cap Y$ :

\bigl{|}\theta(q)\bigr{|}_{h_{Y}}\leq C^{(2)}_{10}|g(w)|^{-1}.

Note that there exists $C^{(2)}_{11}\geq 1$ , such that $(C^{(2)}_{11})^{-1}|g(w)|\leq\mathop{\rm Im}\nolimits g(w)\leq C^{(2)}_{11}|g(w)|$ on $S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\cap\{\arg(g(w))>L^{(0)}/2\}$ . Thus, we obtain the claim of Lemma 5.26. ∎

Lemma 5.27.

There exists an ${\mathbb{R}}_{\geq 0}$ -valued harmonic function $\phi$ on $\{\mathop{\rm Im}\nolimits(g(w))\geq B_{0}\}$ such that the following holds.

•

$\phi=O\bigl{(}|g(w)|^{1-\epsilon_{0}}\bigr{)}$ .
•

For any $Y$ satisfying Condition 5.17, let $s_{Y}$ be the automorphism of $\mathbb{K}_{Y,r}$ determined by $h_{Y}=h_{0|Y}\cdot s_{Y}$ . Then, we obtain $\log\mathop{\rm Tr}\nolimits(s_{Y})\leq\phi$ on $Y\cap\{\mathop{\rm Im}\nolimits(g(w))\geq B_{1}^{(0)}\}$ .

Proof We recall that $\bigl{|}F(h_{0},\theta(q))\bigr{|}_{h_{0}}=O\bigl{(}(\mathop{\rm Im}\nolimits g(w))^{-2}\bigr{)}$ by Lemma 5.25. Hence, there exists an ${\mathbb{R}}_{\geq 0}$ -valued function $\phi_{1}$ on $\{\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ such that (i) $\phi_{1}=O\bigl{(}\log(1+\mathop{\rm Im}\nolimits g(w))\bigr{)}$ , (ii) $\sqrt{-1}\Lambda\overline{\partial}\partial\phi_{1}=\bigl{|}F(h_{0},\theta(q))\bigr{|}$ .

By Corollary 3.6, Lemma 5.15 and Lemma 5.26, there exist $C^{(2)}_{i}$ $(i=20,21)$ , which are independent of $Y$ , such that the following holds on $Y\cap S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\setminus\mathcal{C}_{0}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq-2\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{0})+C^{(2)}_{20}|g(w)|^{\epsilon_{0}}+C^{(2)}_{21}.

(114)

Here, $\theta_{0}>0$ is sufficiently small. There exists $\beta\in{\mathbb{C}}^{\ast}$ such that $\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})>0$ on $\{\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ . Then, there exist $C^{(2)}_{i}$ $(i=22,23)$ which are independent of $Y$ , such that the following inequality holds on $Y\cap S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\setminus\mathcal{C}_{0}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}\leq-2\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{0})+C^{(2)}_{22}\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})+C^{(2)}_{23}.

(115)

Let $\mathcal{D}$ be a connected component of $\mathcal{C}_{0}$ such that

\mathcal{D}\cap S_{0}(g,R^{(0)}_{1},B^{(0)}_{1},L^{(0)}_{1})\neq\emptyset.

Then, $\mathcal{D}$ is relatively compact in $Y\cap S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ . Because $\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}$ is subharmonic on $Y\cap S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ , the estimate (115) holds on $\mathcal{D}$ . Hence, (115) holds on the closure of $Y\cap S_{0}(g,R^{(0)}_{1},B^{(0)}_{1},L_{1}^{(0)})$ . Similarly, there exist $C^{(2)}_{i}>0$ $(i=24,25)$ which are independent of $Y$ such that the following holds on the closure of $Y\cap S_{\pi}(g,R^{(0)}_{1},B_{1}^{(0)},L^{(0)}_{1})$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}\leq-2\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{\pi})+C^{(2)}_{24}\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})+C^{(2)}_{25}.

(116)

Here $\pi-\theta_{\pi}>0$ is sufficiently small.

As in Proposition 3.15, there exists $C^{(2)}_{26}>0$ which is independent of $Y$ such that the following holds on $\bigl{\{}w\in X(R_{0},\delta_{0})\,\big{|}\,\mathop{\rm Im}\nolimits g(w)=B^{(0)}_{1},\,\,|g(w)|\leq 2R^{(0)}_{-1}\bigr{\}}\cap Y$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}\leq C^{(2)}_{26}.

(117)

There exists $\gamma\in{\mathbb{C}}^{\ast}$ such that $\mathop{\rm Re}\nolimits(\gamma g(w)^{1-\epsilon_{0}})>0$ on

\{w\in X(R_{0},\delta_{0})\,|\,\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}.

There exists $C^{(2)}_{27}>0$ which is independent of $Y$ such that the following holds on $Y\cap\{\mathop{\rm Im}\nolimits g(w)=B^{(0)}_{1}\}$ :

\bigl{|}\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{0})\bigr{|}+\bigl{|}\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{\pi})\bigr{|}+\bigl{|}\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})\bigr{|}\leq C^{(2)}_{27}\mathop{\rm Re}\nolimits(\gamma g(w)^{1-\epsilon_{0}}).

(118)

By (115), (116), (117) and (118), there exist $C^{(2)}_{i}>0$ $(i=28,29)$ , which are independent of $Y$ , such that the following holds on $Y\cap\{\mathop{\rm Im}\nolimits g(w)=B^{(0)}_{1}\}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}\leq C^{(2)}_{28}\mathop{\rm Re}\nolimits(\gamma g(w)^{1-\epsilon_{0}})+C^{(2)}_{29}=:\phi_{2}.

We obtain the following on $Y\cap\{\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ :

-\partial_{w}\partial_{\overline{w}}\bigl{(}\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}-\phi_{2}\bigr{)}\leq 0.

Note that $\log\mathop{\rm Tr}\nolimits(s_{Y})=\log r$ on $\partial Y$ , and hence

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}-\phi_{2}\leq\log\mathop{\rm Tr}\nolimits(s_{Y})\leq\log r

on $\partial Y\cap\{\mathop{\rm Im}\nolimits g(w)\geq B^{(0)}_{1}\}$ . We also obtain

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}-\phi_{2}\leq 0\leq\log r

on $Y\cap\{\mathop{\rm Im}\nolimits g(w)=B^{(0)}_{1}\}$ . Hence, we obtain

\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi_{1}-\phi_{2}\leq\log r

on $\partial\bigl{(}Y\cap\{\mathop{\rm Im}\nolimits g(w)\geq B^{(0)}_{1}\}\bigr{)}$ . Hence, we obtain $\log\mathop{\rm Tr}\nolimits(s_{Y})\leq\log r+\phi_{1}+\phi_{2}$ on $Y\cap\{\mathop{\rm Im}\nolimits g(w)\geq B^{(0)}_{1}\}$ . Because $\phi_{1}=O\bigl{(}\log(1+|g(w)|)\bigr{)}$ , there exists $C^{(2)}_{30}>0$ such that $\phi_{1}+\phi_{2}+\log r\leq C^{(2)}_{30}\mathop{\rm Re}\nolimits(\gamma g(w)^{1-\epsilon_{0}})=:\phi$ on $\{\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ . Thus, we obtain Lemma 5.27. ∎

5.6.5. Construction of harmonic metrics

Proposition 5.28.

For any ${\boldsymbol{a}}\in\mathcal{P}$ , there exists $h\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{a}}(h)={\boldsymbol{a}}$ .

Proof For ${\boldsymbol{a}}\in\mathcal{P}$ , we take an auxiliary metric $h_{0}$ as in Lemma 5.25. Let $Y_{1}\subset Y_{2}\subset\cdots$ be a smooth exhaustive family of $X(R_{0},\delta_{0})$ satisfying Condition 5.17. According to Proposition 2.1, we obtain a sequence $h_{Y_{i}}\in\mathop{\rm Harm}\nolimits(q_{|Y_{i}})$ such that $h_{Y_{i}|\partial Y_{i}}=h_{0|\partial Y_{i}}$ . According to Proposition 2.6, by taking a subsequence, we may assume that the sequence $h_{Y_{i}}$ is convergent to $h_{\infty}\in\mathop{\rm Harm}\nolimits(q)$ . Lemma 5.25 and Lemma 5.27 imply ${\boldsymbol{a}}(h_{\infty})={\boldsymbol{a}}$ . Thus, the proof of Proposition 5.28 is completed. ∎

5.6.6. Comparison with harmonic metrics

Let $B_{0}<B^{(0)}_{-1}<B^{(0)}_{0}<B^{(0)}_{1}$ be as in §5.5.4. We can prove the following proposition by the argument of Lemma 5.27.

Proposition 5.29.

There exist a harmonic function $\phi$ on $\{w\in X(R_{0},\delta_{0})\,|\,\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ such that the following holds:

•

$\phi=O\bigl{(}|g(w)|^{1-\epsilon_{0}}\bigr{)}$ .
•

Let $h\in\mathop{\rm Harm}\nolimits(q_{|\{\mathop{\rm Im}\nolimits g(w)>B_{0}\}})$ . Let $Y$ be a relatively compact subset of $X(R_{0},\delta_{0})$ satisfying Condition 5.17. Suppose that $h_{Y}\in\mathop{\rm Harm}\nolimits(q_{|Y})$ satisfies

$h_{Y|\partial Y\cap\{\mathop{\rm Im}\nolimits(g(w))>B_{0}\}}=h_{|\partial Y\cap\{\mathop{\rm Im}\nolimits g(w)>B_{0}\}}.$

Let $s_{Y}$ be the automorphism of $\mathbb{K}_{Y\cap\{\mathop{\rm Im}\nolimits g(w)>B_{0}\},r}$ determined by $h_{Y|Y\cap\{\mathop{\rm Im}\nolimits g(w)>B_{0}\}}=h_{|Y\cap\{\mathop{\rm Im}\nolimits g(w)>B_{0}\}}\cdot s_{Y}$ . Then, we obtain $\log\mathop{\rm Tr}\nolimits(s_{Y})\leq\phi$ on $Y\cap\{\mathop{\rm Im}\nolimits(g(w))>B^{(0)}_{1}\}$ .

Proof In this proof, constants are independent of $h$ and $Y$ . Let $h$ , $h_{Y}\in\mathop{\rm Harm}\nolimits(q_{|Y})$ and $s_{Y}$ be as in the statement. By Proposition 3.7, there exists a constant $C_{1}>0$ such that the following holds on $\{w\in X(R_{0},\delta_{0})\,|\,\mathop{\rm Im}\nolimits g(w)\geq B^{(0)}_{-1}\}$ :

|\theta(q)|_{h}\leq C_{1}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-1}.

By the same argument as the proof of Lemma 5.26, we can prove that there exists a constant $C_{2}>0$ such that the following inequality holds on $Y\cap S_{0}(g,R^{(0)},B^{(0)},L^{(0)})$ :

|\theta(q)|_{h_{Y}}\leq C_{2}\bigl{(}\mathop{\rm Im}\nolimits g(w)\bigr{)}^{-1}.

By Corollary 3.5 and Lemma 5.15, there exist constants $C_{i}>0$ $(i=3,4)$ such that the following inequality holds on $Y\cap S_{0}(g,R^{(0)},B^{(0)},L^{(0)})\setminus\mathcal{C}_{0}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq-2\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{0})+C_{3}\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})+C_{4}.

(119)

Here, $\theta_{0}>0$ is sufficiently small, and $\beta$ is a complex number such that $\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})>0$ on $\{\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ . Note that $\log\mathop{\rm Tr}\nolimits(s_{Y})$ is subharmonic by (21). Hence, the inequality (119) holds on the closure of $Y\cap S_{0}(g,R^{(0)}_{1},B^{(0)}_{1},L^{(0)}_{1})$ . Similarly, there exist constants $C_{i}>0$ $(i=5,6)$ such that the following holds on the closure of $Y\cap S_{\pi}(g,R^{(0)}_{1},B^{(0)}_{1},L^{(0)}_{1})$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq-2\mathop{\rm Re}\nolimits\mathfrak{a}(f,\theta_{\pi})+C_{5}\mathop{\rm Re}\nolimits(\beta g(w)^{\epsilon_{0}})+C_{6}.

(120)

Here, $\pi-\theta_{\pi}>0$ is sufficiently small. As in Proposition 3.15, there exists a constant $C_{7}>0$ such that $\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C_{7}$ on $\{\mathop{\rm Im}\nolimits g(w)=B^{(0)}_{1},\,\,|g(w)|\leq 2R^{(0)}_{-1}\}\cap Y$ . Therefore, there exist positive constants $C_{i}$ $(i=8,9)$ such that the following holds on $Y\cap\{\mathop{\rm Im}\nolimits g(w)=B^{(0)}_{1}\}$ :

\log\mathop{\rm Tr}\nolimits(s_{Y})\leq C_{8}\mathop{\rm Re}\nolimits(\gamma g(w)^{1-\epsilon_{0}})+C_{9}=:\phi_{0}.

Here $\gamma$ is a complex number such that $\mathop{\rm Re}\nolimits(\gamma g(w)^{1-\epsilon_{0}})>0$ on $\{\mathop{\rm Im}\nolimits g(w)\geq B_{0}\}$ .

We set $\phi=\phi_{0}+\log(r)$ . Note that $\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi$ is a subharmonic function. By the construction, we obtain $\log\mathop{\rm Tr}\nolimits(s_{Y})-\phi\leq 0$ on $\partial(Y\cap\{\mathop{\rm Im}\nolimits g(w)\geq B_{1}^{(0)}\})$ , and hence on $Y\cap\{\mathop{\rm Im}\nolimits g(w)\geq B_{1}^{(0)}\}$ . Thus, we obtain Proposition 5.29. ∎

5.6.7. Proof of Theorem 5.3 and Proposition 5.4

We consider the map $\Phi:X(R,\delta)\longrightarrow{\mathbb{C}}$ defined by $w=-\sqrt{-1}\alpha z^{-\rho}$ . If $R$ is sufficiently large and $\delta$ is sufficiently small, we obtain $\Phi:X(R,\delta)\longrightarrow V$ , and it is a holomorphic embedding. By applying Proposition 5.20 and Proposition 5.24 to $\Phi^{\ast}(q)=f\cdot(dw)^{r}$ , we obtain the first two claims of Theorem 5.3.

Let ${\boldsymbol{a}}\in\mathcal{P}$ . By Proposition 5.28, there exists $h_{0,{\boldsymbol{a}}}\in\mathop{\rm Harm}\nolimits(\Phi^{\ast}(q))$ such that ${\boldsymbol{a}}(h_{0,{\boldsymbol{a}}})={\boldsymbol{a}}$ . It induces $h_{0,{\boldsymbol{a}}}\in\mathop{\rm Harm}\nolimits(q_{|\Phi(X(R,\delta))})$ . We extend it to a $G_{r}$ -invariant Hermitian metric of $\mathbb{K}_{V\setminus\varpi^{-1}(0),r}$ such that $\det(h_{1,{\boldsymbol{a}}})=1$ . Let $\{X_{i}\}$ be a smooth exhaustive family of $V$ such that each $\Phi^{-1}(X_{i})$ satisfies Condition 5.17. Let $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ such that $h_{i|\partial X_{i}}=h_{1,{\boldsymbol{a}}|\partial X_{i}}$ . According to Proposition 3.15, we may assume that the sequence $h_{i}$ is convergent, and we obtain $h_{\infty}\in\mathop{\rm Harm}\nolimits(q)$ as the limit of a subsequence of $h_{i}$ . By Proposition 5.29, $h_{\infty}$ satisfies ${\boldsymbol{a}}_{I}(h_{\infty})={\boldsymbol{a}}$ . Thus, we obtain the first claim of Theorem 5.3.

For the proof of Proposition 5.4, we use the notation in §5.5.4. We set $\mathcal{U}_{I,1}=\Phi(\{\mathop{\rm Im}\nolimits(g(w))>B_{0}\})$ and $\mathcal{U}_{I,2}=\Phi(\{\mathop{\rm Im}\nolimits(g(w))>B_{1}^{(0)}\})$ . Let $\{K_{i}\}$ be a smooth exhaustive family of $V\setminus\varpi^{-1}(0)$ such that each $\Phi^{-1}(K_{i})$ satisfies Condition 5.17. Let $\phi_{I}$ denote the function on $\mathcal{U}_{I,1}$ induced by $\phi$ in Proposition 5.29. Then, we obtain Proposition 5.4 from Proposition 5.29. ∎

5.7. Refined estimate in an easy case

Let $f$ be a section of $\mathfrak{B}$ on $V\subset\widetilde{{\mathbb{C}}}$ . Let $I$ be an interval of $V\cap\varpi^{-1}(0)$ such that $\overline{I}\subset V$ and that $I$ is special with respect to $f$ . We assume the following.

Condition 5.30.

There exist a neighbourhood $\mathcal{U}$ of $\overline{I}$ in $V$ , a finite sum $\mathfrak{a}_{I}=\sum_{0<\kappa\leq\rho}\mathfrak{a}_{I,\kappa}z^{\kappa}$ $(\mathfrak{a}_{I,\rho}\neq 0)$ , $\alpha\in{\mathbb{C}}^{\ast}$ , $\ell_{I}\in{\mathbb{R}}$ and $C>1$ such that $\bigl{|}e^{-\mathfrak{a}_{I}}z^{-\ell_{I}}(f-\alpha e^{\mathfrak{a}_{I}}z^{\ell_{I}})\bigr{|}\leq C|z|^{\epsilon}$ for $\epsilon>0$ on $\mathcal{U}\setminus\varpi^{-1}(0)$ .

Recall $\deg(\mathfrak{a}_{I})=\rho$ . We set $q=f\,(dz/z)^{r}$ and

\mathfrak{d}_{I}(q):=\frac{\ell_{I}}{\deg(\mathfrak{a}_{I})}+r.

Let $h\in\mathop{\rm Harm}\nolimits(q)$ . We obtain ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ as in Theorem 5.3. In this easier case, we can obtain the estimate on the behaviour of $h$ up to boundedness. Let ${\boldsymbol{k}}_{I}(h)$ be the tuple of integers obtained from ${\boldsymbol{a}}_{I}(h)$ as in §3.6.1. We shall prove the following proposition in §5.7.1–5.7.2.

Proposition 5.31.

The following estimates holds as $-\mathop{\rm Re}\nolimits(\mathfrak{a}_{I}(z))+\mathfrak{d}_{I}(q)\log|\mathfrak{a}_{I}(z)|\to\infty$ :

\log\Bigl{|}(dz)^{(r+1-2i)/2}\Bigr{|}_{h}-a_{I,i}(h)\Bigl{(}-\mathop{\rm Re}\nolimits(\mathfrak{a}_{I}(z))+\mathfrak{d}_{I}(q)\log|\mathfrak{a}_{I}(z)|\Bigr{)}\\ +(\deg(\mathfrak{a}_{I})+1)\left(\frac{r+1}{2}-i\right)\log|z|\\ -\frac{k_{I,i}}{2}\log\Bigl{(}-\mathop{\rm Re}\nolimits(\mathfrak{a}_{I}(z))+\mathfrak{d}_{I}(f)\log|\mathfrak{a}_{I}(z)|\Bigr{)}=O(1).

(121)

5.7.1. Normalization

Let $\mathcal{U}$ denote an open neighbourhood of $\overline{I}$ . We define the map $F:\mathcal{U}\setminus\varpi^{-1}(0)\longrightarrow{\mathbb{C}}$ by

F(z)=-\sqrt{-1}\Bigl{(}\mathfrak{a}_{I}(z)-\mathfrak{d}_{I}(q)\log\bigl{(}-\sqrt{-1}\mathfrak{a}_{I}(z)\bigr{)}\Bigr{)}.

Let $\varpi_{\infty}:\widetilde{\mathbb{P}}^{1}_{\infty}\longrightarrow\mathbb{P}^{1}$ denote the oriented real blowing up at $\infty$ . We regard ${\mathbb{C}}$ as an open subset of $\widetilde{\mathbb{P}}^{1}_{\infty}$ . Let $w$ denote the standard coordinate system on ${\mathbb{C}}$ . By using the polar decomposition of $w$ , we identify $\widetilde{\mathbb{P}}^{1}_{\infty}\setminus\{0\}$ with $({\mathbb{R}}_{>0}\cup\{\infty\})\times S^{1}$ .

Lemma 5.32.

There exist a neighbourhood $\mathcal{U}^{\prime}$ of $\overline{I}$ in $\mathcal{U}$ and a neighbourhood $\mathcal{V}$ of $\{\infty\}\times[0,\pi]$ in $\widetilde{\mathbb{P}}^{1}_{\infty}$ such that $F$ induces a homeomorphism $\mathcal{U}^{\prime}\simeq\mathcal{V}$ .

Proof It is easy to see that $F$ extends to a continuous map $\mathcal{U}\longrightarrow\widetilde{\mathbb{P}}^{1}_{\infty}$ , which is also denoted by $F$ . Moreover, we can easily check that $F$ induces a homeomorphism of a neighbourhood of $\overline{I}$ in $\varpi^{-1}(0)$ and a neighbourhood of $\{\infty\}\times[0,\pi]$ in $\varpi_{\infty}^{-1}(\infty)$ .

We set $\rho:=\deg(\mathfrak{a}_{I})$ . On a neighbourhood of $\overline{I}$ , we use the real coordinate system given by $(|z|^{\rho},\arg(z))$ . On a neighbourhood of $\{\infty\}\times[0,\pi]$ , we use the real coordinate system given by $(|w|^{-1},\arg(w))$ . Then, it is easy to check that $F$ is of $C^{1}$ -class, and the tangent map at each point of $\overline{I}$ is an isomorphism. Then, the claim follows from the inverse function theorem. ∎

Then, $(F^{-1})^{\ast}(q)$ is expressed as follows:

(F^{-1})^{\ast}(q)=\beta e^{\sqrt{-1}w}(1+\upsilon(w))\,(dw)^{r}.

Here, $\upsilon$ is a holomorphic function satisfying $\upsilon(w)=O(|w|^{-\epsilon})$ for some $\epsilon>0$ , and $\beta$ is a non-zero complex number.

5.7.2. Comparison with the model metric

Let $\delta>0$ and $R>0$ . Let $f_{0}$ be a nowhere vanishing holomorphic function on $X(R,\delta)$ such that $\log|f_{0}|=-\mathop{\rm Im}\nolimits w+O(1)$ . We set $q_{0}:=f_{0}\,(dw)^{r}$ . According to Theorem 5.3, for any $h_{0}\in\mathop{\rm Harm}\nolimits(q_{0})$ , there exist ${\boldsymbol{a}}(h_{0})\in\mathcal{P}$ and $0<\rho<1$ such that the following estimates hold on $\bigl{\{}|w|>R,\,\,\big{|}\arg(w)-\pi/2|<(1-\delta_{1})\pi/2\bigr{\}}$ for any $0<\delta_{1}<1$ :

\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h_{0}}-a_{i}(h_{0})\mathop{\rm Im}\nolimits(w)=O\bigl{(}|w|^{\rho}\bigr{)}.

Note that $\log|f_{0}|=O(1)$ on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,|\mathop{\rm Im}\nolimits(w)|<A\bigr{\}}$ for any $A>0$ in this case. By Corollary 3.8, we obtain that $\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h_{0}}$ is bounded on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,|\mathop{\rm Im}\nolimits(w)|<A\bigr{\}}$ for any $A>0$ . By Proposition 4.4, we obtain the following estimate on $\{w\in X(R+1,\delta)\,|\,\mathop{\rm Im}\nolimits(w)\geq 1\}$ :

\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h_{0}}-a_{i}(h_{0})\mathop{\rm Im}\nolimits(w)=O\bigl{(}\log(1+\mathop{\rm Im}\nolimits(w))\bigr{)}.

Let $f_{1}$ be a holomorphic function on $X(R,\delta)$ such that there exist $C_{1}>0$ and $\kappa>0$ such that

|f_{0}-f_{1}|\leq C_{1}|w|^{-\kappa}|f_{0}|.

We obtain $\log|f_{1}|=-\mathop{\rm Im}\nolimits(w)+O(1)$ . We set $q_{1}:=f_{1}\,(dw)^{r}$ . Similarly, for $h_{1}\in\mathop{\rm Harm}\nolimits(q_{1})$ , there exists ${\boldsymbol{a}}(h_{1})\in\mathcal{P}$ such that the following estimate holds on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,\mathop{\rm Im}\nolimits(w)\geq 1\bigr{\}}$ :

\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h_{1}}-a_{i}(h_{1})\mathop{\rm Im}\nolimits(w)=O\bigl{(}\log(1+\mathop{\rm Im}\nolimits(w))\bigr{)}.

Proposition 5.33.

Suppose that there exist $h_{0}\in\mathop{\rm Harm}\nolimits(q_{0})$ and $h_{1}\in\mathop{\rm Harm}\nolimits(q_{1})$ such that ${\boldsymbol{a}}(h_{0})={\boldsymbol{a}}(h_{1})$ . Then, $h_{0}$ and $h_{1}$ are mutually bounded on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,\mathop{\rm Im}\nolimits(w)\geq 1\bigr{\}}$ .

Proof We set $e_{i}:=(dw)^{(r+1-2i)/2}$ for $i=1,\ldots,r$ . Note that $\theta(q_{1})e_{i}=\theta(q_{0})e_{i}$ $(i=1,\ldots,r)$ , $\theta(q_{1})e_{r}=\theta(q_{0})e_{r}+(f_{1}-f_{0})e_{1}\,dw$ and $\theta(q_{0})(e_{r})=f_{0}\,e_{1}\,dw$ . Because $|f_{1}-f_{0}|\leq C_{1}|w|^{-\kappa}|f_{0}|$ , there exists $C_{2}>0$ such that $|\theta(q_{1})-\theta(q_{0})|_{h_{0}}\leq C_{2}|w|^{-\kappa}|\theta(q_{0})|_{h_{0}}$ . Hence, on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,\mathop{\rm Im}\nolimits(w)\geq 1\bigr{\}}$ , we obtain $|\theta(q_{1})-\theta(q_{0})|_{h_{0}}=O\bigl{(}(1+\mathop{\rm Im}\nolimits(w))^{-1-\kappa}\bigr{)}$ and $|\theta(q_{1})|_{h}=O\bigl{(}(1+\mathop{\rm Im}\nolimits(w))^{-1}\bigr{)}$ . Because $R(h_{0})+\bigl{[}\theta(q_{0}),\theta(q_{0})_{h_{0}}^{\dagger}\bigr{]}=0$ , we obtain

\bigl{|}R(h_{0})+\bigl{[}\theta(q_{1}),\theta(q_{1})_{h_{0}}^{\dagger}\bigr{]}\bigr{|}_{h_{0}}=O\bigl{(}(1+\mathop{\rm Im}\nolimits(w))^{-2-\kappa}\bigr{)}.

Let $s$ be the automorphism of $\mathbb{K}_{X(R,\delta),r}$ determined by $h_{1}=h_{0}s$ . By (21), we obtain the following on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,\mathop{\rm Im}\nolimits(w)\geq 0\bigr{\}}$ :

-2\partial_{z}\partial_{\overline{z}}\log\mathop{\rm Tr}\nolimits(s)\leq\bigl{|}R(h_{0})+\bigl{[}\theta(q_{1}),\theta(q_{1})_{h_{0}}^{\dagger}\bigr{]}\bigr{|}_{h_{0}}=O\bigl{(}(1+\mathop{\rm Im}\nolimits(w))^{-2-\kappa}\bigr{)}.

Note that $\mathop{\rm Tr}\nolimits(s)$ is bounded on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,|\mathop{\rm Im}\nolimits(w)|<A\bigr{\}}$ . By Proposition 4.4, there exists $a\in{\mathbb{R}}$ such that $|\log\mathop{\rm Tr}\nolimits(s)-a\mathop{\rm Im}\nolimits(w)|$ is bounded on $\bigl{\{}w\in X(R+1,\delta)\,\big{|}\,\mathop{\rm Im}\nolimits(w)\geq 0\bigr{\}}$ . Because ${\boldsymbol{a}}(h_{0})={\boldsymbol{a}}(h_{1})$ , we obtain $a=0$ , which implies the boundedness of $\log\mathop{\rm Tr}\nolimits(s)$ . ∎

Let $\upsilon(w)$ be a holomorphic function on $X(R,\delta)$ such that $\upsilon(w)=O(|w|^{-\kappa})$ . Let $\alpha$ be a non-zero complex number. We set $q=\alpha(1+\upsilon)e^{\sqrt{-1}w}(dw)^{r}$ . Let $h\in\mathop{\rm Harm}\nolimits(q)$ . We obtain ${\boldsymbol{a}}(h)\in\mathcal{P}$ and a tuple of integers ${\boldsymbol{k}}(h)$ as in §3.6.1.

Corollary 5.34.

We obtain the following estimates for $i=1,\ldots,r$ on $\bigl{\{}w\in X(R,\delta)\,\big{|}\,\mathop{\rm Im}\nolimits(w)\geq 0\bigr{\}}$ :

\log\bigl{|}(dw)^{(r+1-2i)/2}\bigr{|}_{h}-a_{i}(h)\mathop{\rm Im}\nolimits(w)-\frac{k_{i}(h)}{2}\log\bigl{(}1+\mathop{\rm Im}\nolimits(w)\bigr{)}=O(1).

Proof In the case $\upsilon=0$ , the claim follows from Proposition 3.25 and Theorem 5.3. The general case follows from Proposition 5.33. ∎

We obtain Proposition 5.31 by applying the normalization in §5.7.1 and Corollary 5.34. ∎

6. Global results

6.1. Holomorphic $r$ -differentials with multiple growth orders on a punctured disc

6.1.1. General case

Let $U$ be a neighbourhood of $0$ in ${\mathbb{C}}$ . Let $\varpi:\widetilde{U}\longrightarrow U$ denote the oriented real blowing up. Let $f$ be a section of $\mathfrak{B}_{\widetilde{U}}$ on $\widetilde{U}$ . Let $\mathcal{S}(f)$ denote the set of the intervals which are special with respect to $f$ . For any $I\in\mathcal{S}(f)$ , we fix a branch of $\log z$ around $I$ . Then, $\rho(I)\in{\mathbb{R}}_{>0}$ and $\alpha_{I}\in{\mathbb{C}}^{\ast}$ are determined as

\deg(\mathfrak{a}(f,\theta)-\alpha_{I}z^{-\rho(I)})<\rho(I)

for $\theta\in I\setminus\mathcal{Z}(f)$ . We set $q=f\,(dz)^{r}$ . We set $\mathcal{S}(q):=\mathcal{S}(f)$ . Let $U_{0}\subset U$ be a relatively compact neighbourhood of $0$ in $U$ . We set $U^{\ast}=U\setminus\{0\}$ and $U_{0}^{\ast}=U_{0}\setminus\{0\}$ .

Theorem 6.1.

•

For any $h\in\mathop{\rm Harm}\nolimits(q)$ and for any $I\in\mathcal{S}(q)$ , there exist ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ and $\epsilon>0$ such that the following estimates hold as $|z|\to 0$ on $\bigl{\{}|\arg(\alpha_{I}z^{-\rho(I)})-\pi|<(1-\delta)\pi/2\bigr{\}}$ for any $\delta>0$ :

\log\bigl{|}(dz)^{(r+1)/2-i}\bigr{|}_{h}+a_{i}(h)\mathop{\rm Re}\nolimits\bigl{(}\alpha_{I}z^{-\rho(I)}\bigr{)}=O\bigl{(}|z|^{-\rho(I)+\epsilon}\bigr{)}.

•

If $h_{i}\in\mathop{\rm Harm}\nolimits(q)$ $(i=1,2)$ satisfy ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ for any $I\in\mathcal{S}(q)$ , then $h_{1}$ and $h_{2}$ are mutually bounded on $U_{0}^{\ast}$ .
•

For any $({\boldsymbol{a}}_{I})_{I\in\mathcal{S}(q)}\in\prod_{I\in\mathcal{S}(q)}\mathcal{P}$ , there exists $h\in\mathop{\rm Harm}\nolimits(q)$ such that ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}_{I}$ .

Proof We obtain the first claims from Theorem 5.3. We obtain the second claim from Proposition 5.1, Theorem 5.2 and Theorem 5.3.

Let us prove the third claim. Let $({\boldsymbol{a}}_{I})_{I\in\mathcal{S}(q)}\in\prod_{I\in\mathcal{S}(q)}\mathcal{P}$ . For each $I\in\mathcal{S}(q)$ , let $V_{I}$ be a relatively compact neighbourhood of $\overline{I}$ in $\widetilde{U}$ . There exist relatively compact neighbourhoods $\mathcal{U}_{I,i}$ $(i=1,2)$ , a function $\phi_{I}$ , and a smooth exhaustive family $\{K_{I,i}\}$ , as in Proposition 5.4. By Theorem 5.3, there exists a $G_{r}$ -invariant Hermitian metric $h_{0}$ of $\mathbb{K}_{U\setminus\{0\},r}$ such that $\det(h_{0})=1$ and that $h_{0|\mathcal{U}_{I,1}}\in\mathop{\rm Harm}\nolimits(q_{|\mathcal{U}_{I,1}})$ with ${\boldsymbol{a}}_{I}(h_{0})={\boldsymbol{a}}_{I}$ for $I\in\mathcal{S}(q)$ . Let $\{X_{i}\}$ be a smooth exhaustive family for $U\setminus\{0\}$ such that (i) $\coprod_{I}K_{I,i}\subset X_{i}$ , (ii) $K_{I,i}\cap\mathcal{U}_{I,1}=X_{i}\cap\mathcal{U}_{I,1}$ for any $I$ . We obtain $h_{i}\in\mathop{\rm Harm}\nolimits(q_{|X_{i}})$ such that $h_{i|\partial X_{i}}=h_{0|\partial X_{i}}$ . We may assume that the sequence $\{h_{i}\}$ is convergent, and we obtain $h_{\infty}\in\mathop{\rm Harm}\nolimits(q)$ as the limit. By Proposition 5.4, we obtain ${\boldsymbol{a}}_{I}(h_{\infty})={\boldsymbol{a}}_{I}$ for $I\in\mathcal{S}(q)$ . ∎

6.1.2. Refined estimate in the nowhere vanishing case

Let us state a refined result in the case where $q$ is nowhere vanishing.

Lemma 6.2.

Let $q$ be a holomorphic $r$ -differential with multiple growth orders on $(U,0)$ . If $q$ is nowhere vanishing, then there exist an integer $\ell(q)$ and a meromorphic function $\mathfrak{a}$ on $(U,0)$ such that $q=z^{\ell(q)}e^{\mathfrak{a}(z)}(dz/z)^{r}$ .

Proof We describe $q$ as $q=\beta(dz/z)^{r}$ for a holomorphic function $\beta$ . There exists an integer $\ell$ such that $\mathfrak{a}=\log(z^{-\ell}\beta)$ is well defined as a single-valued holomorphic function on $U^{\ast}$ . Because $\beta$ induces a section of $\mathfrak{B}$ , we obtain $\mathop{\rm Re}\nolimits(\mathfrak{a})=O\bigl{(}|z|^{-\rho}\bigr{)}$ for some $\rho>0$ . Because $\mathop{\rm Re}\nolimits(\mathfrak{a})$ is harmonic, there exists $\rho_{1}>0$ such that $\partial_{x}\mathop{\rm Re}\nolimits(\mathfrak{a})=O(|z|^{-\rho_{1}})$ and $\partial_{y}\mathop{\rm Re}\nolimits(\mathfrak{a})=O(|z|^{-\rho_{1}})$ , where $(x,y)$ is the real coordinate system induced by $z=x+\sqrt{-1}y$ . By the Cauchy-Riemann equation, we obtain that $\partial_{x}\mathop{\rm Im}\nolimits(\mathfrak{a})=O(|z|^{-\rho_{1}})$ and $\partial_{y}\mathop{\rm Im}\nolimits(\mathfrak{a})=O(|z|^{-\rho_{1}})$ . Hence, there exists $\rho_{2}>0$ such that $\mathop{\rm Im}\nolimits(\mathfrak{a})=O(|z|^{-\rho_{2}})$ . Thus, we obtain that $\mathfrak{a}$ is meromorphic at $z=0$ . ∎

For the description $q=z^{\ell(q)}e^{\mathfrak{a}(z)}(dz/z)^{r}$ , we obtain the integer $m(q)$ such that $z^{m(q)}\mathfrak{a}(z)$ is holomorphic at $z=0$ , and $\alpha:=(z^{m(q)}\mathfrak{a}(z))_{z=0}\neq 0$ . We set $\mathfrak{d}(q):=\frac{\ell(q)}{m(q)}+r$ . We also set

T(q):=\{0\leq\theta<2\pi\,|\,-m(q)\theta+\arg(\alpha)\equiv\pi,\mod 2\pi\}.

Note that special intervals with respect to $q$ are $\bigl{\{}|\theta-\theta_{0}|<\pi/2m(q)\bigr{\}}$ for $\theta_{0}\in T(q)$ .

Proposition 6.3.

Let $h\in\mathop{\rm Harm}\nolimits(q)$ . For any $\theta_{0}\in T(q)$ , we obtain the tuple of real numbers

{\boldsymbol{a}}(h,\theta_{0})=(a_{0}(h,\theta_{0}),\ldots,a_{r-1}(h,\theta_{0}))\in\mathcal{P}

determined by the following estimates for $i=1,\ldots,r$ as $-\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(z)\bigr{)}+\mathfrak{d}(q)\log|\mathfrak{a}(z)|\to\infty$ on $\bigl{\{}|\arg(z)-\theta_{0}|<\pi/2m(q)\bigr{\}}$ :

\log\Bigl{|}(dz)^{(r+1)/2-i}\Bigr{|}_{h}-a_{i}(h,\theta_{0})\Bigl{(}-\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(z)\bigr{)}\\ +\mathfrak{d}(q)\log|\mathfrak{a}(z)|\Bigr{)}+(m(q)+1)\left(\frac{r+1}{2}-i\right)\log|z|\\ =O\left(\log\Bigl{(}-\mathop{\rm Re}\nolimits(\mathfrak{a}(z))+\mathfrak{d}(q)\log|\mathfrak{a}(z)|\Bigr{)}\right).

(122)

¿From ${\boldsymbol{a}}(h,\theta_{0})\in\mathcal{P}$ , we determine the integers $k_{i}(h,\theta_{0})$ $(i=0,\ldots,r-1)$ as in §3.6.1. Then, we obtain the following estimates as $-\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(z)\bigr{)}+\mathfrak{d}(q)\log|\mathfrak{a}(z)|\to\infty$ on $\bigl{\{}|\arg(z)-\theta_{0}|<\pi/2m(q)\bigr{\}}$ :

\log\Bigl{|}(dz)^{(r+1)/2-i}\Bigr{|}_{h}-a_{i}(h,\theta_{0})\Bigl{(}-\mathop{\rm Re}\nolimits\bigl{(}\mathfrak{a}(z)\bigr{)}+\mathfrak{d}(q)\log|\mathfrak{a}(z)|\Bigr{)}\\ +(m(q)+1)\left(\frac{r+1}{2}-i\right)\log|z|\\ -\frac{k_{i}(h,\theta_{0})}{2}\log\Bigl{(}-\mathop{\rm Re}\nolimits(\mathfrak{a}(z))+\mathfrak{d}(q)\log|\mathfrak{a}(z)|\Bigr{)}=O\bigl{(}1\bigr{)}.

(123)

Proof It follows from Proposition 5.31. ∎

6.2. The case of punctured Riemann surfaces

6.2.1. Statement

Let $X$ be a Riemann surface. Let $D$ be a finite subset of $X$ . For each $P\in D$ , let $(X_{P},z_{P})$ be a holomorphic coordinate neighbourhood such that $z_{P}(P)=0$ . Set $X_{P}^{\ast}:=X_{P}\setminus\{P\}$ . Let $\varpi_{P}:\widetilde{X}_{P}\to X_{P}$ denote the oriented blow up along $P$ .

Let $q$ be an $r$ -differential on $X$ . We obtain the expression $q_{|X_{P}^{\ast}}=f_{P}\,(dz_{P}/z_{P})^{r}$ . Let $D_{\mathop{\rm mero}\nolimits}$ be the points $P$ of $D$ such that $f_{P}$ are meromorphic at $P$ . Let $D_{>0}$ denote the set of $P$ such that $f_{P}$ is holomorphic at $P$ and that $f_{P}(P)=0$ . We set $D_{\leq 0}:=D_{\mathop{\rm mero}\nolimits}\setminus D_{>0}$ .

For each $P\in D_{>0}$ , we have the description $q_{P}=w_{P}^{m_{P}}f_{P}(dw_{P}/w_{P})^{r}$ , where $f_{P}$ is holomorphic at $P$ such that $f_{P}(P)\neq 0$ . Let $\mathcal{P}(q,P)$ denote the set of ${\boldsymbol{b}}=(b_{1},b_{2},\ldots,b_{r})\in{\mathbb{R}}^{r}$ satisfying

b_{1}\geq b_{2}\geq\cdots\geq b_{r}\geq b_{1}-m_{P},\quad\sum b_{i}+r(r+1)/2=0.

We set $D_{\mathop{\rm ess}\nolimits}:=D\setminus D_{\mathop{\rm mero}\nolimits}$ . We assume that for any $P\in D_{\mathop{\rm ess}\nolimits}$ , $f_{P}$ is a section of $\mathfrak{B}_{\widetilde{X}_{P}}$ . At each $P\in D_{\mathop{\rm ess}\nolimits}$ , we obtain the set of the intervals $\mathcal{S}(q,P)$ in $\varpi_{P}^{-1}(P)$ which are special with respect to $q_{|X_{P}^{\ast}}$ .

Let $h\in\mathop{\rm Harm}\nolimits(q)$ . For any $P\in D_{\mathop{\rm ess}\nolimits}$ and $I\in\mathcal{S}(q,P)$ , we obtain ${\boldsymbol{a}}_{P,I}(h)\in\mathcal{P}$ as in Theorem 6.1. For any $P\in D_{>0}$ , we obtain ${\boldsymbol{b}}(h)\in\mathcal{P}(q,P)$ as in Proposition 3.21. Thus, we obtain the map

\mathop{\rm Harm}\nolimits(q)\longrightarrow\prod_{P\in D_{\mathop{\rm ess}\nolimits}}\prod_{I\in\mathcal{S}(q,P)}\mathcal{P}\times\prod_{P\in D_{>0}}\mathcal{P}(q,P).

(124)

We introduce a boundary condition at infinity of $X$ when $X$ is non-compact. Because $(K_{X\setminus D}^{(r+1-2i)/2})^{-1}\otimes K_{X\setminus D}^{(r+1-2(i+1))/2}\simeq K_{X\setminus D}^{-1}$ , the restrictions of $h_{|K_{X\setminus D}^{(r+1-2i/2)}}$ and $h_{|K_{X\setminus D}^{(r+1-2(i+1))/2}}$ induce a Kähler metric $g(h)_{i}$ of $X\setminus D$ $(i=1,\ldots,r-1)$ .

Definition 6.4.

We say that $h\in\mathop{\rm Harm}\nolimits(q)$ is complete at infinity of $X$ if there exists a relatively compact open neighbourhood $N$ of $D$ such that $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are complete. Let $\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ denote the set of $h\in\mathop{\rm Harm}\nolimits(q)$ which is complete at infinity of $X$ .

Lemma 6.5.

Let $N$ be any relatively compact open neighbourhood of $D$ in $X$ .

•

For any $h\in\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ , $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are complete. Moreover, $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are mutually bounded, and $|q|_{g(h)_{i}|X\setminus N}$ are bounded.
•

Any $h_{j}\in\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ $(j=1,2)$ are mutually bounded on $X\setminus N$ .

Proof The first claim follows from [24, Proposition 3.27]. The second follows from [24, Proposition 3.29]. ∎

We obtain the map

\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)\longrightarrow\prod_{P\in D_{\mathop{\rm ess}\nolimits}}\prod_{I\in\mathcal{S}(q,P)}\mathcal{P}\times\prod_{P\in D_{>0}}\mathcal{P}(q,P).

(125)

We set $\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q;D,\mathop{\rm c}\nolimits):=\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)\cap\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ . Let $\mathcal{P}^{{\mathbb{R}}}(q,P)$ denote the set of ${\boldsymbol{b}}\in\mathcal{P}(q,P)$ such that $b_{i}+b_{r+1-i}=-r-1$ . Let $\mathcal{P}^{{\mathbb{R}}}$ denote the set of ${\boldsymbol{a}}\in\mathcal{P}$ such that $a_{i}+a_{r+1-i}=0$ . As the restriction of (124) and (125), we obtain the following maps:

\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q)\longrightarrow\prod_{P\in D_{\mathop{\rm ess}\nolimits}}\prod_{I\in\mathcal{S}(q,P)}\mathcal{P}^{{\mathbb{R}}}\times\prod_{P\in D_{>0}}\mathcal{P}^{{\mathbb{R}}}(q,P),

(126)

\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q;D,\mathop{\rm c}\nolimits)\longrightarrow\prod_{P\in D_{\mathop{\rm ess}\nolimits}}\prod_{I\in\mathcal{S}(q,P)}\mathcal{P}^{{\mathbb{R}}}\times\prod_{P\in D_{>0}}\mathcal{P}^{{\mathbb{R}}}(q,P).

(127)

Theorem 6.6.

The maps (125) and (127) are bijective. In particular, if $X$ is compact, the maps (124) and (126) are bijective.

The theorem follows from Lemma 6.8, Lemma 6.10 and Lemma 6.11 below.

Remark 6.7.

If the zero set of $q$ is finite, we obtain a refined estimate around each $P\in D_{\mathop{\rm ess}\nolimits}$ as in Proposition 6.3.

6.2.2. Uniqueness

We set $\mathcal{S}(q):=\coprod_{P\in D_{\mathop{\rm ess}\nolimits}}\mathcal{S}(q,P)$ .

Lemma 6.8.

Suppose that $h_{j}\in\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ $(j=1,2)$ satisfy ${\boldsymbol{b}}_{P}(h_{1})={\boldsymbol{b}}_{P}(h_{2})$ for any $P\in D_{>0}$ and ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ for $I\in\mathcal{S}(q)$ . Then, we obtain $h_{1}=h_{2}$ .

Proof As in Theorem 1.4, there uniquely exists $h^{\mathop{\rm c}\nolimits}\in\mathop{\rm Harm}\nolimits(q)$ such that $g(h^{\mathop{\rm c}\nolimits})_{i}$ are complete on $X\setminus D$ . Recall that $g(h^{\mathop{\rm c}\nolimits})_{i}$ are mutually bounded, and that the Gaussian curvature of $g(h^{\mathop{\rm c}\nolimits})_{i}$ are bounded from below [24, Lemma 3.15].

Let $g$ be a Kähler metric of $X$ such that (i) the Gaussian curvature is bounded from below, (ii) the following condition is satisfied for a relatively compact neighbourhood $N$ of $D$ .

Condition 6.9.

$g_{|X\setminus N}$ is mutually bounded with the metrics $g(h^{\mathop{\rm c}\nolimits})_{i}$ $(i=1,\ldots,r-1)$ .

Let $s$ be the automorphism of $\mathbb{K}_{X\setminus D,r}$ determined by $h_{2}=h_{1}\cdot s$ . Let $\Lambda$ denote the adjoint of the multiplication of the Kähler form associated with $g$ . By (20), we obtain the following on $X\setminus D$ :

\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s)=-\bigl{|}[\theta(q),s]s^{-1/2}\bigr{|}^{2}_{h_{1},g}-\bigl{|}\overline{\partial}_{E}(s)s^{-1/2}\bigr{|}^{2}_{h_{1},g}\\ \leq-\bigl{|}[\theta(q),s]s^{-1/2}\bigr{|}^{2}_{h_{1},g}.

(128)

By Proposition 3.19, Proposition 3.21, Theorem 6.1, and Lemma 6.5, we obtain the boundedness of $\mathop{\rm Tr}\nolimits(s)$ . Hence, the inequality (128) holds across $D$ . (See [39, Lemma 2.2].) It particularly implies that $\mathop{\rm Tr}\nolimits(s)$ is subharmonic on $X$ .

Let $N$ be a relatively compact neighbourhood of $D$ . If

\sup_{Q\in X\setminus D}\mathop{\rm Tr}\nolimits(s)(Q)=\sup_{Q\in N\setminus D}\mathop{\rm Tr}\nolimits(s)(Q),

the maximum principle for subharmonic functions implies that $\mathop{\rm Tr}\nolimits(s)$ is constant on $X\setminus D$ , and hence $[\theta(q),s]=0$ on $X\setminus D$ . Together with $\det(s)=1$ , we obtain $s=\mathop{\rm id}\nolimits$ .

Let us study the case $\sup_{Q\in X\setminus D}\mathop{\rm Tr}\nolimits(s)(Q)>\sup_{Q\in N\setminus D}\mathop{\rm Tr}\nolimits(s)(Q)$ . By Omori-Yau maximum principle [24, Lemma 3.2], there exists a sequence $Q_{\ell}\in X\setminus N$ such that

-2\sqrt{-1}\Lambda\overline{\partial}\partial\mathop{\rm Tr}\nolimits(s)(Q_{\ell})\leq\ell^{-1},\quad\mathop{\rm Tr}\nolimits(s)(Q_{\ell})>\sup_{Q\in X\setminus D}\mathop{\rm Tr}\nolimits(s)(Q_{\ell})-\ell^{-1}.

By (128), there exists $C_{1}>0$ such that the following holds for any $\ell$ :

\bigl{|}[s,\theta(q)]s^{-1/2}\bigr{|}^{2}_{h_{1},g}(Q_{\ell})\leq C_{1}\ell^{-1}.

Let $z_{\ell}$ be a holomorphic coordinate around $Q_{\ell}$ such that $|dz_{\ell}|^{2}_{g}(Q_{\ell})=2$ . By Condition 6.9, there exists $B>0$ such that $\bigl{|}(dz_{\ell})^{(r+1-2i)/2}\bigr{|}_{h_{1}}\cdot\bigl{|}(dz_{\ell})^{(r+1-2(i+1))/2}\bigr{|}_{h_{1}}^{-1}\leq B$ for any $\ell$ . Hence, by [24, Lemma 3.9] there exists $C_{2}>0$ and $\ell_{2}$ such that $\mathop{\rm Tr}\nolimits(s)(Q_{\ell})\leq r(1+C_{2}\ell^{-1/2})$ for any $\ell>\ell_{2}$ . Hence, we obtain $\sup_{Q\in X\setminus D}\mathop{\rm Tr}\nolimits(s)(Q)\leq r$ . Because $\det(s)=1$ , we obtain $\mathop{\rm Tr}\nolimits(s)\geq r$ for any $Q$ . Therefore, $\mathop{\rm Tr}\nolimits(s)$ is constantly $r$ , and we obtain $s=\mathop{\rm id}\nolimits$ . ∎

Lemma 6.10.

Suppose that $h\in\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ satisfies ${\boldsymbol{b}}_{P}(h)\in\mathcal{P}^{{\mathbb{R}}}(q,P)$ for any $P\in D_{>0}$ and ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}^{{\mathbb{R}}}$ for any $I\in\mathcal{S}(q)$ . Then, we obtain $h\in\mathop{\rm Harm}\nolimits^{{\mathbb{R}}}(q;D,\mathop{\rm c}\nolimits)$ .

Proof We obtain $h^{\lor}\in\mathop{\rm Harm}\nolimits(q)$ as the dual of $h$ by using the natural identification of $\mathbb{K}_{X\setminus D,r}$ with its dual. Clearly, $h^{\lor}$ is also complete at infinity of $X$ . Because ${\boldsymbol{b}}_{P}(h)\in\mathcal{P}^{{\mathbb{R}}}(q,P)$ $(P\in D_{>0})$ and ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}^{{\mathbb{R}}}$ $(I\in\mathcal{S}(q))$ , we obtain ${\boldsymbol{b}}_{P}(h)={\boldsymbol{b}}_{P}(h^{\lor})$ and ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}_{I}(h^{\lor})$ . By Lemma 6.8, we obtain $h=h^{\lor}$ . ∎

6.2.3. Existence

Lemma 6.11.

For any ${\boldsymbol{b}}_{P}\in\mathcal{P}(q,P)$ $(P\in D_{>0})$ and ${\boldsymbol{a}}_{I}\in\mathcal{P}$ $(I\in\mathcal{S}(q))$ , there exists $h\in\mathop{\rm Harm}\nolimits(q;D,\mathop{\rm c}\nolimits)$ such that ${\boldsymbol{b}}_{P}(h)={\boldsymbol{b}}_{P}$ for any $P\in D_{>0}$ and that ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}_{I}$ for any $I\in\mathcal{S}(q)$ .

Proof Let $h^{\mathop{\rm c}\nolimits}$ be as in the proof of Lemma 6.8. For any $({\boldsymbol{a}}_{I})_{I\in\mathcal{S}(q)}$ and $({\boldsymbol{b}}_{P})_{P\in D_{>0}}$ , by Proposition 3.23 and Theorem 6.1, there exists a $G_{r}$ -invariant Hermitian metric $h_{0}$ of $\mathbb{K}_{X\setminus D,r}$ such that the following holds.

•

There exists a relatively compact neighbourhood $N_{1}$ of $D$ such that $h_{0|X\setminus N_{1}}=h^{\mathop{\rm c}\nolimits}_{|X\setminus N_{1}}$ .
•

There exists a relatively compact neighbourhood $N_{2}\subset N_{1}$ of $D$ such that $h_{0|N_{2}\setminus D}\in\mathop{\rm Harm}\nolimits(q_{|N_{2}\setminus D})$ with ${\boldsymbol{b}}_{P}(h_{0})={\boldsymbol{b}}_{P}$ for $P\in D_{>0}$ , and ${\boldsymbol{a}}_{I}(h_{0})={\boldsymbol{a}}_{I}$ for $I\in\mathcal{S}(q)$ .
•

$\det(h_{0})=1$ .

Then, according to Proposition 3.18, there exists $h\in\mathop{\rm Harm}\nolimits(q;D,h^{\mathop{\rm c}\nolimits})$ which is mutually bounded with $h_{0}$ . Thus, we obtain Lemma 6.11, and the proof of Theorem 6.6 is completed. ∎

6.3. Examples on ${\mathbb{C}}$

Let $\varpi_{\infty}:\widetilde{\mathbb{P}}^{1}_{\infty}\longrightarrow\mathbb{P}^{1}$ be the oriented real blow up at $\infty$ . We identify $\varpi_{\infty}^{-1}(\infty)$ with $S^{1}$ by the polar decomposition $z=|z|e^{\sqrt{-1}\theta}$ .

6.3.1. Classification in the case of $\gamma(z)e^{\mathfrak{a}(z)}(dz)^{r}$

Let $\rho$ be a positive integer. Let $\alpha$ be a non-zero complex number. Let $\Lambda(\alpha,\rho)$ denote the set of the connected components of

\bigl{\{}e^{\sqrt{-1}\theta}\in S^{1}\,\big{|}\,\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\rho\theta})<0\bigr{\}}.

Note that $|\Lambda(\alpha,\rho)|=\rho$ . Let $\mathfrak{a}$ be a polynomial of the form

\mathfrak{a}=\alpha z^{\rho}+\sum_{j=1}^{\rho-1}\mathfrak{a}_{j}z^{j}.

Let $\gamma(z)$ be a non-zero polynomial. We set $q=\gamma(z)e^{\mathfrak{a}}(dz)^{r}$ . Clearly, $\Lambda(\alpha,\rho)$ is the set of the intervals special with respect to $q$ . For any $h\in\mathop{\rm Harm}\nolimits(q)$ , we obtain ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ $(I\in\Lambda(\alpha,\rho))$ . The following proposition is a special case of Theorem 6.6.

Proposition 6.12.

The map $\mathop{\rm Harm}\nolimits(q)\longrightarrow\mathcal{P}^{\Lambda(\alpha,\rho)}$ is bijective. ∎

Remark 6.13.

We can obtain a refined estimate around $\infty$ as in Proposition 6.3.

6.3.2. The case of $\sum_{i=1}^{n}\gamma_{i}(z)e^{\mathfrak{a}_{i}(z)}\,(dz)^{r}$

Let $n\geq 2$ . Let $\mathfrak{a}_{i}(z)$ $(i=1,2,\ldots,n)$ be mutually distinct polynomials such that $\mathfrak{a}_{i}(0)=0$ . If $\mathfrak{a}_{i}\neq 0$ , we set $\rho_{i}:=\deg(\mathfrak{a}_{i})$ , and let $\alpha_{i}\neq 0$ denote the coefficients of the top term of $\mathfrak{a}_{i}$ , i.e.,

\mathfrak{a}_{i}(z)=\alpha_{i}z^{\rho_{i}}+\sum_{j=1}^{\rho_{i}-1}\mathfrak{a}_{i,j}z^{j}.

If $\mathfrak{a}_{i}=0$ , we set $\rho_{i}=0$ and $\alpha_{i}=0$ . Let $\gamma_{i}(z)$ $(i=1,\ldots,n)$ be non-zero polynomials. We set $q=\sum_{i=1}^{n}\gamma_{i}(z)e^{\mathfrak{a}_{i}(z)}(dz)^{r}$ . The following proposition is a special case of Theorem 6.6.

Proposition 6.14.

•

Suppose that there exist $\rho>0$ and $\alpha\in{\mathbb{C}}^{\ast}$ such that $\rho_{i}=\rho$ and $\alpha_{i}/\alpha\in{\mathbb{R}}_{>0}$ for any $i$ . Then, $\Lambda(\alpha,\rho)$ is the set of the intervals which are special with respect to $q$ . Hence, for any $h\in\mathop{\rm Harm}\nolimits(q)$ , we obtain ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ $(I\in\Lambda(\alpha,\rho))$ , which induces a bijection $\mathop{\rm Harm}\nolimits(q)\longrightarrow\mathcal{P}^{\Lambda(\alpha,\rho)}$ .
•

Otherwise, we obtain $\mathop{\rm Harm}\nolimits(q)=\{h^{\mathop{\rm c}\nolimits}(q)\}$ .

Proof Let us prove the first claim. We set $\kappa_{0}:=\max\{\alpha_{i}/\alpha\}$ and $\kappa_{1}:=\min\{\alpha_{i}/\alpha\}$ . Let $f$ be the entire function determined by $q=f(dz)^{r}$ . We set $\mathcal{Z}(f)=\bigl{\{}\theta\in S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}\,\big{|}\,\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\theta})=0\bigr{\}}$ . Let $\theta\in S^{1}\setminus\mathcal{Z}(q)$ . There exists $\mathfrak{a}_{i}$ such that $\mathfrak{a}_{i}=\mathfrak{a}(f,\theta)$ (See Definition 4.17 for $\mathfrak{a}(f,\theta)$ .) Note that $\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\rho\theta})\neq 0$ . If $\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\rho\theta})>0$ , we obtain $\alpha_{i}/\alpha=\kappa_{0}$ . If $\mathop{\rm Re}\nolimits(\alpha e^{\sqrt{-1}\rho\theta})<0$ , we obtain $\alpha_{i}/\alpha=\kappa_{1}$ . Then, the first claim is clear.

Let us prove the second claim. Note that $q$ is not meromorphic at $\infty$ under the assumption that $n\geq 2$ . Suppose that there exists a special interval $I\subset S^{1}$ with respect to $f$ . There exists $\rho>0$ such that the length of $I$ is $\pi/\rho$ . Because there exists $\mathfrak{a}_{i}$ such that $\rho=\rho_{i}$ , $\rho$ is a positive integer. Suppose that there exists $\mathfrak{a}_{j}$ such that $\rho_{j}>\rho$ . Then, there exists $\theta\in I\setminus\mathcal{Z}(f)$ such that $\mathop{\rm Re}\nolimits(\alpha_{j}e^{\sqrt{-1}\rho_{j}\theta})>0$ . It contradicts that $I$ is negative. Suppose that there exists $\rho_{j}<\rho$ . For any $\theta\in I\setminus\mathcal{Z}(f)$ , there exists $\mathfrak{a}_{i}$ such that $\rho_{i}=\rho$ and $\mathfrak{a}_{i}=\mathfrak{a}(f,\theta)$ . But, we obtain $\mathop{\rm Re}\nolimits(\alpha_{j}s^{\rho_{j}}e^{\sqrt{-1}\rho_{j}\theta})>\mathop{\rm Re}\nolimits(\alpha_{i}s^{\rho_{i}}e^{\sqrt{-1}\rho_{i}\theta})$ for any sufficiently large $s$ , which contradicts $\mathfrak{a}_{i}=\mathfrak{a}(f,\theta)$ . Therefore, we obtain $\rho_{i}=\rho$ for any $i$ . There exists $\alpha\in{\mathbb{C}}^{\ast}$ such that for any $\theta\in I\setminus\mathcal{Z}(f)$ we obtain $\deg(\mathfrak{a}(f,\theta)-\alpha z^{\rho})<\rho$ . Suppose that there exists $\mathfrak{a}_{j}$ such that $\alpha_{j}/\alpha\not\in{\mathbb{R}}_{>0}$ . Then, there exists $\theta\in I$ such that $\mathop{\rm Re}\nolimits(\alpha_{j}e^{\sqrt{-1}\rho\theta})>0$ . It contradicts that $I$ is negative. Hence, we obtain $\alpha_{j}/\alpha\in{\mathbb{R}}_{>0}$ for any $j$ . ∎

Corollary 6.15.

Let $P$ be any non-zero polynomial. and let $Q$ be any non-constant polynomial. We set $q_{\cos}=P\cos(Q)(dz)^{r}$ , $q_{\sin}=P\sin(Q)(dz)^{r}$ , $q_{\cosh}=P\cosh(Q)(dz)^{r}$ and $q_{\sinh}=P\sinh(Q)(dz)^{r}$ . Because $P\cos(Q)=(Pe^{\sqrt{-1}Q}+Pe^{-\sqrt{-1}Q})/2$ , we obtain $\mathop{\rm Harm}\nolimits(q_{\cos})=\{h^{\mathop{\rm c}\nolimits}(q_{\cos})\}$ . Similarly, we obtain $\mathop{\rm Harm}\nolimits(q_{\sin})=\{h^{\mathop{\rm c}\nolimits}(q_{\sin})\}$ , $\mathop{\rm Harm}\nolimits(q_{\cosh})=\{h^{\mathop{\rm c}\nolimits}(q_{\cosh})\}$ and $\mathop{\rm Harm}\nolimits(q_{\sinh})=\{h^{\mathop{\rm c}\nolimits}(q_{\sinh})\}$ . ∎

6.3.3. Airy function

Let $\Gamma$ be a continuous map ${\mathbb{R}}\longrightarrow{\mathbb{C}}$ such that $\Gamma(t)=t(1+\sqrt{-3})/2$ for $t<-1$ and $\Gamma(t)=t(-1+\sqrt{-3})/2$ for $t>1$ . Recall that an Airy function is defined as

\mathop{\rm Ai}\nolimits(z):=\int_{\Gamma}e^{zt-t^{3}/3}dt.

(129)

(For example, see [47, §22].) It is an entire function. Recall that it is a solution of the linear differential equation $\partial_{z}^{2}u-zu=0$ , and hence it induces a section of $\mathfrak{B}_{\widetilde{\mathbb{P}}^{1}}$ . On $|\arg(z)|<\pi$ , it satisfies the following estimate (for example, see [47, §23]):

e^{\frac{2}{3}z^{3/2}}z^{1/4}\Bigl{(}\mathop{\rm Ai}\nolimits(z)-\frac{z^{-1/4}}{2\sqrt{\pi}}e^{-\frac{2}{3}z^{3/2}}\Bigr{)}=O(|z|^{-3/2}).

(130)

Let $\gamma(z)$ be any non-zero polynomial. We set $q=\gamma(z)\mathop{\rm Ai}\nolimits(z)\,(dz)^{r}$ . Because of (130), the special interval with respect to $q$ is $I=\{-\pi/3<\theta<\pi/3\}$ . We obtain the following proposition from Theorem 6.6.

Proposition 6.16.

For any $h\in\mathop{\rm Harm}\nolimits(q)$ , we obtain ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ , which induces a bijection $\mathop{\rm Harm}\nolimits(q)\longrightarrow\mathcal{P}$ . ∎

Remark 6.17.

More generally, let $f$ be a solution of a differential equation $(\partial_{z}^{n}+\sum_{j=0}^{n-1}a_{j}(z)\partial_{z}^{j})f=0$ , where $a_{j}(z)$ are polynomials. Then, $f$ induces a section of $\mathfrak{B}_{\widetilde{\mathbb{P}}^{1}_{\infty}}$ according to the classical asymptotic analysis. (For example, see [28, §II.1].) Hence, Theorem 6.6 allows us to classify $\mathop{\rm Harm}\nolimits(f\,(dz)^{r})$ in terms of the asymptotic expansion of $f$ .

7. A generalization

7.1. Preliminary

7.1.1. Meromorphic extensions

Let $X$ be a Riemann surface with a discrete subset $S$ . Let $\mathcal{O}_{X}(\ast S)$ denote the sheaf of meromorphic functions on $X$ which may have poles along $S$ . Let $\iota:X\setminus S\longrightarrow X$ denote the inclusion. For any locally free $\mathcal{O}_{X\setminus S}$ -module $\mathcal{M}$ , we obtain the $\iota_{\ast}\mathcal{O}_{X\setminus S}$ -module $\iota_{\ast}\mathcal{M}$ .

Definition 7.1.

A meromorphic extension of $\mathcal{M}$ on $(X,S)$ is a locally free $\mathcal{O}_{X}(\ast S)$ -submodule $\widetilde{\mathcal{M}}\subset\iota_{\ast}\mathcal{M}$ such that $\iota^{\ast}\widetilde{\mathcal{M}}=\mathcal{M}$ .

7.1.2. Meromorphic extensions of cyclic Higgs bundles on a punctured disc

Let $U$ be a neighbourhood of $0$ in ${\mathbb{C}}$ . We set $U^{\circ}:=U\setminus\{0\}$ . Let $\mathfrak{L}_{i}$ $(i=1,\ldots r)$ be locally free $\mathcal{O}_{U^{\circ}}$ -modules of rank one. Let $\psi_{i}:\mathfrak{L}_{i}\longrightarrow\mathfrak{L}_{i+1}\otimes K_{U^{\circ}}$ $(i=1,\ldots,r-1)$ and $\psi_{r}:\mathfrak{L}_{r}\longrightarrow\mathfrak{L}_{1}\otimes K_{U^{\circ}}$ be $\mathcal{O}_{U^{\circ}}$ -morphisms. We obtain a holomorphic section $q_{\leq r-1}:=\psi_{r-1}\circ\cdots\circ\psi_{1}$ of $\mathop{\rm Hom}\nolimits(\mathfrak{L}_{1},\mathfrak{L}_{r})\otimes K_{U^{\circ}}^{\otimes(r-1)}$ . We assume the following.

•

The zero set of $q_{\leq r-1}$ is finite.

We set $\mathfrak{V}:=\bigoplus_{i=1}^{r}\mathfrak{L}_{i}$ . Let $\theta$ be the Higgs field of $\mathfrak{V}$ induced by $\psi_{i}$ $(i=1\ldots,r)$ .

Proposition 7.2.

For any meromorphic extension $\widetilde{\det(\mathfrak{V})}$ of $\det(\mathfrak{V})$ on $(U,0)$ , there uniquely exist meromorphic extensions $\widetilde{\mathfrak{L}}_{i}$ of $\mathfrak{L}_{i}$ $(i=1,\ldots,r)$ on $(U,0)$ such that (i) $\theta(\widetilde{\mathfrak{L}}_{i})\subset\widetilde{\mathfrak{L}}_{i+1}\otimes K_{U}$ $(i=1,\ldots,r-1)$ , (ii) $\det(\widetilde{\mathfrak{V}})=\widetilde{\det(\mathfrak{V})}$ , where $\widetilde{\mathfrak{V}}=\bigoplus\widetilde{\mathfrak{L}}_{i}$ .

Proof By shrinking $U$ , we may assume that $q_{\leq r-1}$ is nowhere vanishing on $U^{\circ}$ . We obtain the morphisms $\beta_{i}:\mathfrak{L}_{i}\longrightarrow\mathfrak{L}_{i+1}$ $(i=1,\ldots,r-1)$ by $\psi_{i}=\beta_{i}\,(dz/z)$ .

Because $U^{\circ}$ is Stein, there exists a global frame $u^{\star}_{1}$ of $\mathfrak{L}_{1}$ . We set $u^{\star}_{i}:=\beta_{i-1}\circ\cdots\beta_{1}(u^{\star}_{1})$ on $U^{\circ}$ for $i=2,\ldots,r$ . We obtain the meromorphic extensions $\widetilde{\mathfrak{L}}^{\star}_{i}=\mathcal{O}_{U}\cdot u^{\star}_{i}$ of $\mathfrak{L}_{i}$ .

Let $v_{0}$ be a frame of $\widetilde{\det(\mathfrak{V})}$ . Let $f$ be the holomorphic function on $U^{\circ}$ determined by $u^{\star}_{1}\wedge\cdots\wedge u^{\star}_{r}=f\cdot v_{0}$ . There exist an integer $\ell$ and a holomorphic function $g$ on $U^{\circ}$ such that $f=z^{\ell}e^{g}$ . We set $u_{i}:=e^{-g/r}u_{i}^{\star}$ and we obtain meromorphic extensions $\widetilde{\mathfrak{L}}_{i}:=\mathcal{O}_{U}(\ast 0)u_{i}$ . We set $\widetilde{\mathfrak{V}}=\bigoplus\widetilde{\mathfrak{L}}_{i}$ . Then, by the construction, we obtain $\theta(\widetilde{\mathfrak{L}}_{i})\subset\widetilde{\mathfrak{L}}_{i+1}\otimes K_{U}$ for $i=1,\ldots,r-1$ , and $\det(\widetilde{\mathfrak{V}})=\widetilde{\det(\mathfrak{V})}$ .

Let $\widetilde{\mathfrak{L}}^{\sharp}_{i}$ be meromorphic extensions of $\mathfrak{L}_{i}$ such that (i) $\theta\widetilde{\mathfrak{L}}^{\sharp}_{i}\subset\widetilde{\mathfrak{L}}^{\sharp}_{i+1}\otimes K_{U}$ $(i=1,\ldots,r-1)$ , (ii) $\det(\widetilde{\mathfrak{V}}^{\sharp})=\widetilde{\det(\mathfrak{V})}=\det(\widetilde{\mathfrak{V}})$ , where $\widetilde{\mathfrak{V}}^{\sharp}=\bigoplus\widetilde{\mathfrak{L}}^{\sharp}_{i}$ . There exist frames $u^{\sharp}_{i}$ of $\widetilde{\mathfrak{L}}^{\sharp}_{i}$ such that $\theta(u^{\sharp}_{i})=u^{\sharp}_{i+1}dz/z$ for $i=1,\ldots,r-1$ .

Let $\gamma$ be the holomorphic function on $U^{\circ}$ determined by $u^{\sharp}_{1}=\gamma u_{1}$ . Then, we obtain $u^{\sharp}_{i}=\gamma u_{i}$ for $i=1,\ldots,r$ . Because both $u^{\sharp}_{1}\wedge\cdots\wedge u^{\sharp}_{r}$ and $u_{1}\wedge\cdots\wedge u_{r}$ are sections of $\widetilde{\det(\mathfrak{V})}$ , $\gamma^{r}$ is meromorphic at $0$ . Hence, we obtain that $\gamma$ is meromorphic, i.e., $\widetilde{\mathfrak{L}}_{i}=\widetilde{\mathfrak{L}}_{i}^{\sharp}$ . ∎

7.2. Cyclic Higgs bundles

Let $X$ be a Riemann surface with a finite subset $D\subset X$ . Assume that $X\setminus D$ is an open Riemann surface, i.e., $X$ is open, or $X$ is compact and $D\neq\emptyset$ . (See Remark 7.12.) For each $P\in D$ , let $(X_{P},z_{P})$ be a holomorphic coordinate neighbourhood such that $z_{P}(P)=0$ . Set $X_{P}^{\ast}:=X_{P}\setminus\{P\}$ .

Let $r\geq 2$ . Let $L_{i}$ $(i=1,\ldots,r)$ be holomorphic line bundles on $X\setminus D$ . Let $\psi_{i}:L_{i}\longrightarrow L_{i+1}\otimes K_{X\setminus D}$ and $\psi_{r}:L_{r}\longrightarrow L_{1}\otimes K_{X\setminus D}$ be non-zero morphisms. We set $E:=\bigoplus L_{i}$ . Let $\theta$ be the cyclic Higgs field of $E$ induced by $\psi_{i}$ $(i=1,\ldots,r)$ .

We obtain the holomorphic section $q:=\psi_{r}\circ\cdots\circ\psi_{1}=(-1)^{r-1}\det(\theta)$ of $K_{X\setminus D}^{\otimes r}$ , and the holomorphic section $q_{\leq r-1}:=\psi_{r-1}\circ\cdots\circ\psi_{1}$ of $\mathop{\rm Hom}\nolimits(L_{1},L_{r})\otimes K_{X\setminus D}^{\otimes(r-1)}$ . We assume the following.

•

The zero set of $q_{\leq r-1}$ is finite.
•

$q$ is not constantly $0$ .
•

Let $f_{P}$ $(P\in D)$ be holomorphic functions on $X_{P}^{\ast}$ obtained as $q_{|X_{P}^{\ast}}=f_{P}\,(dz_{P}/z_{P})^{r}$ . Then, $f_{P}$ have multiple growth orders at $P$ .

Let $D_{\mathop{\rm mero}\nolimits}$ denote the set of $P\in D$ such that $f_{P}$ is meromorphic at $P$ . We put $D_{\mathop{\rm ess}\nolimits}:=D\setminus D_{\mathop{\rm mero}\nolimits}$ . For $P\in D_{\mathop{\rm mero}\nolimits}$ , we describe $f_{P}=z_{P}^{m_{P}}\beta_{P}$ , where $\beta_{P}$ is a nowhere vanishing holomorphic function on $X_{P}$ . Let $D_{>0}$ denote the set of $P\in D_{\mathop{\rm mero}\nolimits}$ such that $m_{P}>0$ . We set $D_{\leq 0}:=D_{\mathop{\rm mero}\nolimits}\setminus D_{>0}$ .

7.2.1. Flat metrics on the determinant and local frames

Let $h_{\det(E)}$ be a Hermitian metric of $\det(E)$ such that the Chern connection of $\bigl{(}\det(E),h_{\det(E)}\bigr{)}$ is flat. Note that such a metric exists because $\det(E)\simeq\mathcal{O}_{X\setminus D}$ under the assumption that $X\setminus D$ is non-compact. (For example, see [12, Theorem 30.3].)

Lemma 7.3.

For each $P\in D$ , there exist frames ${\boldsymbol{v}}_{P}=(v_{P,i}\,|\,i=1,\ldots,r)$ of $L_{i|X_{P}^{\ast}}$ and a real number $c({\boldsymbol{v}}_{P})$ such that the following condition is satisfied.

•

$\theta(v_{P,i})=v_{P,i+1}(dz_{P}/z_{P})$ for $i=1,\ldots,r-1$ .
•

We set $\omega_{P}=v_{P,1}\wedge\cdots\wedge v_{P,r}$ . Then, we obtain $|z_{P}|^{c({\boldsymbol{v}}_{P})}\cdot|\omega_{P}|_{h_{\det(E)}}=1$ .

Proof There exists a frame $w_{P}$ of $\det(E)_{|X_{P}^{\ast}}$ such that $|w_{P}|_{h_{\det(E)}}=|z_{P}|^{d_{P}}$ for a real number $d_{P}$ . By Proposition 7.2, there exists a frame $v^{\prime}_{P,i}$ of $L_{i|X_{P}^{\ast}}$ $(i=1,\ldots,r)$ such that $\theta(v^{\prime}_{P,i})=v^{\prime}_{P,i+1}(dz_{P}/z_{P})$ for $i=1,\ldots,r-1$ , and $v^{\prime}_{P,1}\wedge\cdots\wedge v^{\prime}_{P,r}=z_{P}^{n_{P}}\beta_{P}\cdot w_{P}$ for an integer $n_{P}$ and a nowhere vanishing holomorphic function $\beta_{P}$ on $X_{P}$ . By fixing an $r$ -th root $\beta_{P}^{1/r}$ of $\beta_{P}$ , and by setting $v_{P,i}:=\beta_{P}^{-1/r}v_{P,i}^{\prime}$ , we obtain the claim of the lemma. ∎

For $P\in D_{>0}$ , let $\mathcal{P}(q,P,{\boldsymbol{v}}_{P})$ denote the set of ${\boldsymbol{b}}=(b_{i})\in{\mathbb{R}}^{r}$ satisfying the following conditions:

\sum_{i=1}^{r}b_{i}=c({\boldsymbol{v}}_{P}),\quad b_{i}\geq b_{i+1}\,\,(i=1,\ldots,r-1),\quad b_{r}\geq b_{1}-m_{P}.

Remark 7.4.

If $(E,\theta)=(\mathbb{K}_{X\setminus D,r},\theta(q))$ for an $r$ -differential $q$ , we usually choose $h_{\det(E)}=1$ and

v_{P,i}=z_{P}^{i}(dz_{P})^{(r+1-2i)/2}\quad(i=1,\ldots,r),

for which we obtain $c({\boldsymbol{v}}_{P})=-r(r+1)/2$ .

7.2.2. Harmonic metrics and the associated parabolic weights

We consider the $G_{r}$ -action on $L_{i}$ by $a\bullet u_{i}=a^{i}u_{i}$ , which induces a $G_{r}$ -action on $E$ . For any open subset $Y\subset X\setminus D$ , let $\mathop{\rm Harm}\nolimits^{G_{r}}((E,\theta)_{|Y},h_{\det(E)})$ denote the set of $G_{r}$ -invariant harmonic metrics $h$ of $(E,\theta)_{|Y}$ such that $\det(h)=h_{\det(E)|Y}$ .

Proposition 7.5.

Let $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ .

•

For any $P\in D_{>0}$ , there exists ${\boldsymbol{b}}_{P}(h)\in\mathcal{P}(q,P,{\boldsymbol{v}}_{P})$ determined by the following condition.

$b_{P,i}(h)=\inf\Bigl{\{}b\in{\mathbb{R}}\,\Big{|}\,\mbox{\rm$|z_{P}|^{b}|v_{P,i}|_{h}$ is bounded}\Bigr{\}}.$
•

For any $P\in D_{\mathop{\rm ess}\nolimits}$ and any $I\in\mathcal{S}(q,P)$ , there exist ${\boldsymbol{a}}_{I}(h)\in\mathcal{P}$ and $\epsilon>0$ such that the following estimates hold as $|z_{P}|\to 0$ on $\bigl{\{}|\arg(\alpha_{I}z_{P}^{-\rho(I)})-\pi|<(1-\delta)\pi/2\bigr{\}}$ for any $\delta>0$ :

$\log|v_{i}|_{h}+a_{I,i}(h)\mathop{\rm Re}\nolimits(\alpha_{I}z_{P}^{-\rho(I)})=O\bigl{(}|z_{P}|^{-\rho(I)+\epsilon}\bigr{)}$

Here, $\alpha_{I}$ and $\rho(I)$ are determined for $I$ and $q=(-1)^{r-1}\det(\theta)$ as in §6.1.1.

Proof It is enough to study the case $D=\{P\}$ and $X=X_{P}$ . There exists a holomorphic line bundle $\det(E)^{1/r}$ on $X_{P}^{\ast}$ with an isomorphism

(\det(E)^{1/r})^{\otimes r}\simeq\det(E).

(131)

There exists a flat metric $h_{\det(E)^{1/r}}$ of $\det(E)^{1/r}$ such that $h_{\det(E)^{1/r}}^{\otimes\,r}=h_{\det(E)}$ under the isomorphism (131). By using the frames ${\boldsymbol{v}}_{P}$ and $z_{P}^{i}(dz_{P})^{(r+1-2i)/2}$ , we obtain an isomorphism $(E,\theta)\otimes\det(E)^{-1/r}\simeq(\mathbb{K}_{X_{P}^{\ast},r},\theta(q))$ . Then, the claims of the proposition are reduced to Proposition 3.21 and Theorem 6.1. ∎

Proposition 7.6.

Suppose that $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ satisfy ${\boldsymbol{b}}_{P}(h_{1})={\boldsymbol{b}}_{P}(h_{2})$ for any $P\in D_{>0}$ and ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ for any $I\in\mathcal{S}(q)$ . Then, for any relatively compact neighbourhood $N$ of $D$ in $X$ , $h_{1}$ and $h_{2}$ are mutually bounded on $N\setminus D$ .

Proof By using the local isomorphisms in the proof of Proposition 7.5, the claim is reduced to Proposition 3.19, Proposition 3.21 and Theorem 6.1. ∎

7.2.3. Completeness at infinity

Let $Z(q_{\leq r-1})$ denote the zero set of $q_{\leq r-1}$ , which is assumed to be finite. We set $\widetilde{D}=D\cup Z(q_{\leq r-1})$ . Note that $\psi_{i}$ $(i=1,\ldots,r-1)$ induce isomorphisms $(L_{i+1}/L_{i})_{|X\setminus\widetilde{D}}\simeq K_{X\setminus\widetilde{D}}^{-1}$ . Hence, for any $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ , the Hermitian metric $h_{|L_{i+1}}\otimes h_{|L_{i}}^{-1}$ $(i=1,\ldots,r-1)$ on $L_{i+1}/L_{i}$ induce Kähler metrics $g(h)_{i}$ on $X\setminus\widetilde{D}$ .

Definition 7.7.

$h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ is called complete at infinity of $X$ if there exists a relatively compact open subset $N$ of $\widetilde{D}$ such that $g(h)_{i|X\setminus N}$ are complete. Let $\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)$ denote the set of $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ which are complete at infinity of $X$ .

Because $X\setminus\widetilde{D}$ is an open Riemann surface, there exists an isomorphism $\det(E)_{|X\setminus\widetilde{D}}\simeq\mathcal{O}_{X\setminus\widetilde{D}}$ . Hence, there exists a holomorphic line bundle $\det(E)^{1/r}$ on $X\setminus\widetilde{D}$ with an isomorphism $\rho:(\det(E)^{1/r})^{\otimes r}\simeq\det(E)_{|X\setminus\widetilde{D}}$ . We say that such $(\det(E)^{1/r}_{1},\rho_{1})$ and $(\det(E)^{1/r}_{2},\rho_{2})$ are isomorphic if there exists an isomorphism $\kappa:\det(E)^{1/r}_{1}\simeq\det(E)^{1/r}_{2}$ such that $\rho_{2}\circ\kappa^{\otimes\,r}=\rho_{1}$ .

Lemma 7.8.

If we choose $(\det(E)^{1/r},\rho)$ appropriately, there exist isomorphisms $\varphi_{i}:L_{i|X\setminus\widetilde{D}}\simeq K_{X\setminus\widetilde{D}}^{(r+1-2i)/2}\otimes\det(E)^{1/r}$ $(i=1,\ldots,r)$ such that the following conditions are satisfied:

•

The following diagram is commutative.

\begin{CD}E_{|X\setminus\widetilde{D}}@>{\bigoplus\varphi_{i}}>{}>\mathbb{K}_{X\setminus\widetilde{D},r}\otimes\det(E)^{1/r}\\ @V{\theta}V{}V@V{\theta(q)}V{}V\\ E_{|X\setminus\widetilde{D}}\otimes K_{X\setminus\widetilde{D}}@>{\bigoplus\varphi_{i}}>{}>\mathbb{K}_{X\setminus\widetilde{D},r}\otimes\det(E)^{1/r}\otimes K_{X\setminus\widetilde{D}}.\end{CD}

•

The induced morphism

\det\Bigl{(}\bigoplus\varphi_{i}\Bigr{)}:\det(E_{|X\setminus\widetilde{D}})\simeq\det\bigl{(}\mathbb{K}_{X\setminus\widetilde{D},r}\otimes\det(E)^{1/r}\bigr{)}=(\det(E)^{1/r})^{\otimes\,r}

is equal to $\rho^{-1}$ .

Such $(\det(E)^{1/r},\rho)$ is unique up to isomorphisms. Such an isomorphism $\bigoplus\varphi_{i}$ is unique up to the multiplication of an $r$ -th root of $1$ .

Proof We take a holomorphic line bundle $\det(E)_{0}^{1/r}$ with an isomorphism $\rho_{0}:(\det(E)_{0}^{1/r})^{\otimes r}\simeq\det(E)$ . There exists an isomorphism of holomorphic line bundles $\nu_{1}:L_{1|X\setminus\widetilde{D}}\simeq K_{X\setminus\widetilde{D}}^{\otimes(r-1)/2}\otimes\det(E)^{1/r}_{0|X\setminus\widetilde{D}}$ . We obtain the holomorphic isomorphisms $\nu_{i}:L_{i|X\setminus\widetilde{D}}\simeq K_{X\setminus\widetilde{D}}^{\otimes(r+1-2i)/2}\otimes\det(E)^{1/r}_{0|X\setminus\widetilde{D}}$ $(i=2,\ldots,r)$ such that the following diagram is commutative:

\begin{CD}\bigoplus_{i=1}^{r}L_{i|X\setminus\widetilde{D}}@>{\bigoplus\nu_{i}}>{}>\mathbb{K}_{X\setminus\widetilde{D},r}\otimes\det(E)^{1/r}_{0}\\ @V{\theta}V{}V@V{\theta(q)}V{}V\\ \bigoplus_{i=1}^{r}L_{i|X\setminus\widetilde{D}}\otimes K_{X\setminus\widetilde{D}}@>{\bigoplus\nu_{i}}>{}>\mathbb{K}_{X\setminus\widetilde{D},r}\otimes\det(E)_{0}^{1/r}\otimes K_{X\setminus\widetilde{D}}.\end{CD}

The induced isomorphism $\det\bigl{(}\bigoplus\nu_{i}\bigr{)}:\det(E)_{|X\setminus\widetilde{D}}\simeq(\det(E)_{0}^{1/r})^{\otimes r}$ is equal to $\beta\rho_{0}^{-1}$ , where $\beta$ is a nowhere vanishing holomorphic function.

Let $X\setminus\widetilde{D}=\bigcup_{\lambda\in\Lambda}U_{\lambda}$ be a locally finite covering by simply connected open subsets $U_{\lambda}$ . There exists an $r$ -th root $\beta^{1/r}_{\lambda}$ of $\beta_{|U_{\lambda}}$ . If $U_{\lambda}\cap U_{\mu}\neq\emptyset$ , then $\gamma_{\lambda,\mu}:=\beta^{1/r}_{\lambda}\cdot(\beta^{1/r}_{\mu})^{-1}$ induces a locally constant function $U_{\lambda}\cap U_{\mu}\longrightarrow G_{r}\subset{\mathbb{C}}^{\ast}$ . We obtain $\gamma_{\lambda,\mu}\cdot\gamma_{\mu,\lambda}=1$ . If $U_{\lambda}\cap U_{\mu}\cap U_{\nu}\neq\emptyset$ , then $\gamma_{\lambda,\mu}\cdot\gamma_{\mu,\nu}\cdot\gamma_{\nu,\lambda}=1$ . We obtain a holomorphic line bundle $\mathcal{L}$ on $X\setminus\widetilde{D}$ by gluing holomorphic line bundles $\mathcal{O}_{U_{\lambda}}\cdot e_{\lambda}$ $(\lambda\in\Lambda)$ on $U_{\lambda}$ via the relation $e_{\lambda}=\gamma_{\lambda,\mu}e_{\mu}$ on $U_{\lambda}\cap U_{\mu}$ . Because $\gamma_{\lambda,\mu}^{r}=1$ , we obtain an isomorphism $\rho_{\mathcal{L}}:\mathcal{L}^{\otimes\,r}\simeq\mathcal{O}_{X\setminus\widetilde{D}}$ by $e_{\lambda}^{\otimes r}\longmapsto 1$ .

For each $\lambda$ , we obtain an isomorphism $\varphi_{i,\lambda}:=\beta_{\lambda}^{-1/r}\nu_{i|U_{\lambda}}\otimes e_{\lambda}:L_{i|U_{\lambda}}\simeq\bigl{(}K_{X\setminus\widetilde{D}}^{(r+1-2i)/2}\otimes\det(E)^{1/r}_{0}\bigr{)}_{|U_{\lambda}}\otimes\mathcal{L}_{|U_{\lambda}}$ . By the construction, the morphisms $\varphi_{i,\lambda}$ $(\lambda\in\Lambda)$ determine an isomorphism $\varphi_{i}:L_{i|X\setminus\widetilde{D}}\simeq K_{X\setminus\widetilde{D}}^{(r+1-2i)/2}\otimes\det(E)_{0}^{1/r}\otimes\mathcal{L}$ . By setting $(\det(E)^{1/r},\rho)=(\det(E)_{0}^{1/r}\otimes\mathcal{L},\rho_{0}\otimes\rho_{\mathcal{L}})$ , we obtain the claim of the lemma. ∎

In the following, we choose $(\det(E)^{1/r},\rho)$ as in Lemma 7.8. Let $h_{\det(E)^{1/r}}$ be the flat metric of $\det(E)^{1/r}$ determined by $h_{\det(E)^{1/r}}^{\otimes r}=h_{\det(E)|X\setminus\widetilde{D}}$ . We obtain the bijection

\Upsilon:\mathop{\rm Harm}\nolimits(q_{|X\setminus\widetilde{D}})\simeq\mathop{\rm Harm}\nolimits^{G_{r}}\bigl{(}(E,\theta)_{|X\setminus\widetilde{D}},h_{\det(E)|X\setminus\widetilde{D}}\bigr{)}

determined by $\Upsilon(h)=h\otimes h_{\det(E)^{1/r}}$ under the isomorphism in Lemma 7.8.

Proposition 7.9.

Let $N$ be any relatively compact open neighbourhood of $\widetilde{D}$ in $X$ .

•

For any $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)$ , the restrictions $h_{1|X\setminus N}$ and $h_{2|X\setminus N}$ are mutually bounded.
•

For any $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)$ , the metrics $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are mutually bounded, and $|q_{|X\setminus N}|_{g(h)_{i}}$ are bounded.

Proof We may assume that the boundary of $N$ in $X$ is smooth and compact. We obtain the first claim by applying [24, Proposition 3.29] to $\Upsilon^{-1}(h_{i})_{|X\setminus N}$ . For any $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ , we obtain $g(h)_{i|X\setminus\widetilde{D}}=g(\Upsilon^{-1}(h_{|X\setminus\widetilde{D}}))_{i}$ $(i=1,\ldots,r-1)$ . Hence, we obtain the second claim from [24, Proposition 3.27]. ∎

Note that we may naturally regard $|\psi_{i}|_{h}^{2}$ as real sections of $K_{X}\otimes\overline{K_{X}}$ . Note that $g(h)_{i}=|\psi_{i}|_{h}^{2}$ $(i=1,\ldots,r-1)$ . The second claim of Proposition 7.9 is reworded as follows.

Corollary 7.10.

Let $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)$ . For any relatively compact open neighbourhood $N$ of $\widetilde{D}$ in $X$ , the metrics $(|\psi_{i}|_{h}^{2})_{|X\setminus N}$ $(i=1,\ldots,r-1)$ are mutually bounded, and $\bigl{(}|\psi_{r}|_{h}^{2}/|\psi_{i}|_{h}^{2}\bigr{)}_{|X\setminus N}$ $(i=1,\ldots,r-1)$ are bounded. ∎

7.2.4. Existence and uniqueness

By Proposition 7.5, we obtain the following map:

\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}(q,P,{\boldsymbol{v}}_{P})\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}.

(132)

We obtain the following map as the restriction of (132):

\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)\longrightarrow\prod_{P\in D_{>0}}\mathcal{P}(q,P,{\boldsymbol{v}}_{P})\times\prod_{I\in\mathcal{S}(q)}\mathcal{P}.

(133)

Theorem 7.11.

The map (133) is a bijection. In particular, if $X$ is compact, the map (132) is a bijection.

Proof Suppose that $h_{1},h_{2}\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)};D,\mathop{\rm c}\nolimits)$ satisfy ${\boldsymbol{b}}_{P}(h_{1})={\boldsymbol{b}}_{P}(h_{2})$ for any $P\in D_{>0}$ and ${\boldsymbol{a}}_{I}(h_{1})={\boldsymbol{a}}_{I}(h_{2})$ for any $I\in\mathcal{S}(q)$ . By Proposition 7.6 and Proposition 7.9, we obtain that $h_{1}$ and $h_{2}$ are mutually bounded. By applying the argument in the proof of Theorem 6.6, we obtain $h_{1}=h_{2}$ .

Let ${\boldsymbol{b}}_{P}$ $(P\in D_{>0})$ and ${\boldsymbol{a}}_{I}$ $(I\in\mathcal{S}(q))$ . Let $N$ be a relatively compact open neighbourhood of $D$ in $X\setminus Z(q_{\leq r-1})$ . By using the argument in the proof of Proposition 7.5, we can construct

h_{N}\in\mathop{\rm Harm}\nolimits((E,\theta,h_{\det(E)})_{|X\setminus Z(q_{\leq r-1})})

such that ${\boldsymbol{b}}_{P}(h_{N})={\boldsymbol{b}}_{P}$ for $P\in D_{>0}$ and ${\boldsymbol{a}}_{I}(h_{N})={\boldsymbol{a}}_{I}$ for $I\in\mathcal{S}(q)$ . Let $N_{1}$ be a relatively compact neighbourhood of $D$ in $N$ . Let $N_{2}$ be a relatively compact neighbourhood of $Z(q_{\leq r-1})$ . We set $h^{\mathop{\rm c}\nolimits}_{E,\theta}:=\Upsilon(h^{\mathop{\rm c}\nolimits})$ , where $h^{\mathop{\rm c}\nolimits}$ denotes the unique complete metric in $\mathop{\rm Harm}\nolimits(q_{|X\setminus\widetilde{D}})$ . (See §7.2.3 for $\Upsilon$ .) Then, there exists a $G_{r}$ -invariant Hermitian metric $h_{0}$ of $E$ such that (i) $h_{0|N_{1}\setminus D}=h_{N|\setminus N_{1}\setminus D}$ , (ii) $h_{0|X\setminus(N\cup N_{2})}=h^{\mathop{\rm c}\nolimits}_{E,\theta|X\setminus(N\cup N_{2})}$ , (iii) $\det(h_{0})=h_{\det(E)}$ . By Proposition 3.18, we obtain $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ which is mutually bounded with $h_{0}$ , which satisfies ${\boldsymbol{b}}_{P}(h)={\boldsymbol{b}}_{P}$ $(P\in D_{>0})$ and ${\boldsymbol{a}}_{I}(h)={\boldsymbol{a}}_{I}$ $(I\in\mathcal{S}(q))$ . ∎

Remark 7.12.

This type of classification is well known in the case where $X$ is compact and $D$ is empty. Indeed, under the assumption that $q\neq 0$ , $(E,\theta)$ is stable with respect to the $G_{r}$ -action. Hence, for a prescribed Hermitian metric $h_{\det(E)}$ of $\det(E)$ , there uniquely exists a $G_{r}$ -invariant Hermitian metric $h$ of $E$ such that (i) $h$ is Hermitian-Einstein in the sense that the trace-free part of $F(h,\theta)^{\bot}$ is $0$ , (ii) $\det(h)=h_{\det(E)}$ . If moreover $c_{1}(E)=0$ , we may choose a flat metric $h_{\det(E)}$ of $\det(E)$ , then such $h$ is a harmonic metric.

We obtain the following theorem as a consequence of Theorem 7.11 and Corollary 7.10.

Theorem 7.13.

Suppose $D$ is empty. Let $N\subset X$ be a relatively compact open set containing all zeros of $q_{\leq r-1}$ . There uniquely exists a metric $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ such that the metrics $|\psi_{i}|_{h}^{2}$ $(i=1,\cdots,r-1)$ are complete on $X\setminus N$ . Moreover, on $X\setminus N$ , $|\psi_{i}|^{2}_{h}$ $(i=1,\ldots,r-1)$ are mutually bounded, and $\frac{|\psi_{r}|^{2}_{h}}{|\psi_{i}|^{2}_{h}}$ $(i=1,\cdots,r-1)$ are bounded. ∎

7.2.5. Boundedness at infinity

Let $g$ be a complete Kähler metric of $X$ . Let $N$ be a relatively compact open neighbourhood of $\widetilde{D}$ . We say that $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ is bounded at infinity of $X$ with respect to $g$ if there exists $0<\delta=\delta(N)<1$ such that on $X\setminus N$

\delta\leq|\psi_{i}|_{h,g}^{2}\leq\delta^{-1}\quad(i=1,\ldots,r-1).

(134)

In other words, $g(h)_{i}$ $(i=1,\ldots,r-1)$ are mutually bounded with $g$ on $X\setminus N$ .

Lemma 7.14.

Suppose that there exists $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ which is bounded at infinity of $X$ . Then, $h$ is complete at infinity of $X$ . Moreover, $|q|_{g}$ is bounded on $X\setminus N$ , and hence $|\psi_{r}|_{h,g}$ is bounded on $X\setminus N$ .

Proof The first claim is clear by definition. The boundedness of $|q|_{g}$ on $X\setminus N$ follows from Proposition 7.9. ∎

We obtain the following uniqueness up to boundedness.

Corollary 7.15.

Suppose that $h_{i}\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ $(i=1,2)$ are bounded at infinity of $X$ . Then, $h_{1}$ and $h_{2}$ are mutually bounded on $X\setminus N$ . If moreover $D$ is empty, we obtain $h_{1}=h_{2}$ on $X$ .

Proof The first claim follows from Proposition 7.9 and Lemma 7.14. The second claim follows from Theorem 7.13 and Lemma 7.14. ∎

A bounded solution does not necessarily exist for a prescribed complete Kähler metric.

Corollary 7.16.

Let $g^{\prime}$ be another complete Kähler metric of $X$ . If there exist $h,h^{\prime}\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ such that $h$ (resp. $h^{\prime}$ ) is bounded at infinity of $X$ with respect to $g$ (resp. $g^{\prime}$ ), then $g$ and $g^{\prime}$ are mutually bounded on $X$ .

Proof By Proposition 7.9 and Lemma 7.14, $h$ and $h^{\prime}$ are mutually bounded on $X\setminus N$ . It implies that $|g(h)_{i}|^{2}$ $(i=1,\ldots,r-1)$ are mutually bounded with $g$ and $g^{\prime}$ on $X\setminus N$ . Hence, we obtain that $g$ and $g^{\prime}$ are mutually bounded on $X$ . ∎

Proposition 7.17.

Let $g$ be a complete hyperbolic metric of $X$ . Assume that $|q|_{g}$ is bounded on $X\setminus N$ . Then, $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ is bounded at infinity of $X$ with respect to $g$ if and only if $h$ is complete at infinity.

Proof The “only if” part of the claim is clear by definition. Let us prove that the “if” part of the claim. Let $K$ be a compact neighbourhood of $\widetilde{D}$ in $N$ with smooth boundary. Note that $X\setminus K$ is hyperbolic.

Lemma 7.18.

Let $g_{X\setminus K}$ be a complete hyperbolic metric of $X\setminus K$ .

•

$g\leq g_{X\setminus K}$ on $X\setminus K$ .
•

$g$ and $g_{X\setminus K}$ are mutually bounded on $X\setminus N$ .

Proof We may assume that the Gaussian curvature of $g_{X\setminus K}$ and $g$ are constantly $-2$ . Recall that they correspond to complete solutions of the Toda equation (1) with $r=2$ and $q=0$ on $X\setminus K$ and $X$ , respectively. (See also §1.1.5.) The first claim follows from [24, Theorem 1.7]. The second claim of the lemma follows from [24, Proposition 3.29]. ∎

Because $|q|_{g}$ is bounded on $X$ , $|q_{|X\setminus K}|_{g_{X\setminus K}}$ is bounded. Let $h^{\mathop{\rm c}\nolimits}_{X\setminus K}\in\mathop{\rm Harm}\nolimits(q_{|X\setminus K})$ be the complete solution on $X\setminus K$ . According to [24, Theorem 1.8], $g(h^{\mathop{\rm c}\nolimits}_{X\setminus K})_{i}$ $(i=1,\ldots,r-1)$ are mutually bounded with $g_{X\setminus K}$ . Hence, $g(h^{\mathop{\rm c}\nolimits}_{X\setminus K})_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are mutually bounded with $g_{|X\setminus N}$ .

Suppose that $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ is complete at infinity. By [24, Proposition 3.29], $h^{\mathop{\rm c}\nolimits}_{X\setminus K|X\setminus N}$ and $\Upsilon^{-1}(h_{|X\setminus N})$ are mutually bounded. (See §7.2.3 for $\Upsilon$ .) Hence, $g(h)_{i|X\setminus N}$ $(i=1,\ldots,r-1)$ are mutually bounded with $g_{|X\setminus N}$ , i.e., $h$ is bounded at infinity with respect to $g$ . ∎

We obtain the following theorem as a consequence of Theorem 7.13, Lemma 7.14, and Proposition 7.17.

Theorem 7.19.

Suppose that $D$ is empty. Let $g$ be a complete hyperbolic metric of $X$ . If $q$ is bounded with respect to $g$ , there uniquely exists a bounded solution $h\in\mathop{\rm Harm}\nolimits^{G_{r}}(E,\theta,h_{\det(E)})$ . Conversely, if there exists such a bounded solution $h$ , $q$ is bounded with respect to $g$ . ∎

Remark 7.20.

According to Proposition 7.17, if $g$ is complete hyperbolic and if $|q|_{g}$ is bounded on $X\setminus N$ , we may replace the condition “completeness at infinity” in Theorem 7.11 with the condition “boundedness at infinity with respect to $g$ ”.

8. Appendix

8.1. The case of trivial $r$ -differentials

8.1.1. Rank $2$ case

Let $X$ be a Riemann surface with a finite subset $D$ . Suppose that $X\setminus D$ has a Kähler metric $g_{X\setminus D}$ with infinite volume such that the Gaussian curvature of the Riemannian metric $\mathop{\rm Re}\nolimits(g_{X\setminus D})$ is constantly $-k$ $(k>0)$ .

Proposition 8.1.

For any ${\boldsymbol{a}}=(a_{P})_{P\in D}\in{\mathbb{R}}^{D}_{\geq 0}$ , there exists a Kähler metric $g_{{\boldsymbol{a}}}$ of $X\setminus D$ whose Gaussian curvature is constantly $-k$ , such that the following holds.

•

For any neighbourhood $N$ of $D$ , $g_{{\boldsymbol{a}}|X\setminus N}$ and $g_{X\setminus D|X\setminus N}$ are mutually bounded.
•

Let $(X_{P},z_{P})$ be a coordinate neighbourhood around $P$ such that $z_{P}(P)=0$ . If $a_{P}>0$ , then $g_{{\boldsymbol{a}}|X_{P}\setminus\{P\}}|z_{P}|^{-2(a_{P}-1)}$ is mutually bounded with $dz_{P}\,d\overline{z}_{P}$ around $P$ . If $a_{P}=0$ , then $g_{{\boldsymbol{a}}|X_{P}\setminus\{P\}}$ is mutually bounded with $|z_{P}|^{-2}(\log|z_{P}|)^{-2}dz_{P}\,d\overline{z}_{P}$ around $P$ .

Proof It is enough to study the case where $k=2$ . Recall that in this paper, for a Kähler metric $g$ on $X\setminus D$ , we use the Hermitian metric on $K_{X}^{\ell/2}$ denoted as $g^{-\ell/2}$ , determined as follows.

•

If $g=g_{0}\,dz\otimes d\overline{z}$ for a local holomorphic coordinate $z$ , then $|(dz)^{\ell/2}|^{2}_{g^{-\ell/2}}=(\frac{g_{0}}{2})^{-\ell/2}$ .

Let $\omega_{g}$ denote the associated Kähler form, which is locally $\frac{\sqrt{-1}}{2}g_{0}dz\wedge d\overline{z}$ .

We set $\mathbb{K}_{X\setminus D,2}:=K_{X\setminus D}^{1/2}\oplus K_{X\setminus D}^{-1/2}$ . Let $0_{X\setminus D,2}$ denote the quadratic differential on $X\setminus D$ which is constantly $0$ . It induces the Higgs field $\theta(0_{X\setminus D,2})$ of $\mathbb{K}_{X\setminus D,2}$ . We consider the action of $G_{2}=\{\pm 1\}$ on $K_{X\setminus D}^{\pm 1/2}$ given by $a\bullet x=a^{\pm 1}x$ , which induces a $G_{2}$ -action on $\mathbb{K}_{X\setminus D,2}$ . A Kähler metric $g$ of $X\setminus D$ induces a Hermitian metric $h^{(1)}(g)=g^{-1/2}\oplus g^{1/2}$ .

Lemma 8.2.

$h^{(1)}(g)$ is a harmonic metric if and only if the Gaussian curvature of $\mathop{\rm Re}\nolimits(g)$ is constantly $-2$ .

Proof Let $R(g)$ denote the curvature of the Chern connection of $(K^{-1}_{X\setminus D},g)$ . Then, by a direct calculation, we can check that $h^{(1)}$ is harmonic if and only if $\sqrt{-1}R(g)+2\omega_{g}=0$ , which implies that the Gaussian curvature of $g$ is $-2$ . ∎

Let $\mathop{\rm Harm}\nolimits(0_{X\setminus D,2})$ denote the set of $G_{2}$ -invariant harmonic metrics $h$ of $(\mathbb{K}_{X\setminus D,2},\theta(0_{X\setminus D,2}))$ such that $\det(h)=1$ . Note that $h$ is decomposed as $h=h_{|K_{X\setminus D}^{1/2}}\oplus h_{|K_{X\setminus D}^{-1/2}}$ , and it is equal to $h^{(1)}(g)$ for a Kähler metric $g$ of $X\setminus D$ whose Gaussian curvature is $-2$ .

Let us give some preliminaries on hyperbolic metrics on $X_{P}^{\ast}:=X_{P}\setminus\{P\}$ $(P\in D)$ . We may assume that $X_{P}=\{|z_{P}|<1/2\}$ . It is well known and easy to check that $g_{P,1}:=2(1-|z_{P}|^{2})^{-2}dz_{P}\,d\overline{z}_{P}$ is a hyperbolic metric of $X_{P}$ normalized as $\sqrt{-1}R(g_{P,1})+2\omega_{g_{P,1}}=0$ . For $a>0$ , we set as follows on $X_{P}^{\ast}$ :

g_{P,a}=\frac{2dz_{P}^{a}\,d\overline{z}_{P}^{a}}{(1-|z_{P}|^{2a})^{2}}=\frac{2a^{2}|z_{P}|^{2(a-1)}\,dz_{P}\,d\overline{z}_{P}}{(1-|z_{P}|^{2a})^{2}}.

Because $g_{P,a}$ is locally obtained as the pull back of $g_{P,1}$ by the map $z_{P}\longmapsto z_{P}^{a}$ , it is a hyperbolic metric on $X_{P}^{\ast}$ normalized as $\sqrt{-1}R(g_{P,a})+2\omega_{g_{P,a}}=0$ . We also set

g_{P,0}:=\frac{2dz_{P}\,d\overline{z}_{P}}{|z_{P}|^{2}(\log|z_{P}|^{2})^{2}}.

It is easy to check $\sqrt{-1}R(g_{P,0})+2\omega_{g_{P,0}}=0$ .

Lemma 8.3.

Let $g_{P}$ be any hyperbolic metric of $X_{P}^{\ast}$ normalized as $\sqrt{-1}R(g_{P})+2\omega_{g_{P}}=0$ .

•

There exists $a\geq 0$ such that $g_{P}$ and $g_{P,a}$ are mutually bounded around $P$ .
•

Let $X_{P,1}$ be a relatively compact neighbourhood of $P$ in $X_{P}$ . Then, the volume of $X_{P,1}\setminus\{P\}$ is finite with respect to $g_{P}$ .

Proof Let $h\in\mathop{\rm Harm}\nolimits(0_{X_{P}^{\ast},2})$ . Because $\theta(0_{X_{P}^{\ast},2})$ is nilpotent, the harmonic bundle $(\mathbb{K}_{X_{P}^{\ast},2},\theta(0_{X_{P}^{\ast},2}),h)$ is tame. As in the proof of Proposition 3.21, we obtain the locally free $\mathcal{O}_{X_{P}}$ -modules

\mathcal{P}^{h}_{a}\mathbb{K}_{X_{P}^{\ast},2}=\mathcal{P}^{h}_{a}K^{1/2}_{X_{P}^{\ast}}\oplus\mathcal{P}^{h}_{a}K^{-1/2}_{X_{P}^{\ast}}\quad(a\in{\mathbb{R}}),

and we can prove that there exists $b\in{\mathbb{R}}$ such that $z_{P}(dz_{P})^{1/2}$ and $z_{P}^{2}(dz_{P})^{-1/2}$ are sections of $\mathcal{P}^{h}_{b}\mathbb{K}_{X_{P}^{\ast},2}$ . We obtain the numbers

b_{i}(h):=\inf\bigl{\{}b\in{\mathbb{R}}\,\big{|}\,\mbox{$|z_{P}|^{b}\cdot|z_{P}^{i}(dz_{P})^{(3-2i)/2}|_{h}$ is bounded}\bigr{\}}\quad(i=1,2).

Because $\theta(0_{X_{P}^{\ast},2})$ is nilpotent, we obtain

\theta(0_{X_{P}^{\ast},2})=O\bigl{(}\bigl{|}\log|z_{P}|\bigr{|}^{-1}\bigr{)}dz_{P}/z_{P}

with respect to $h$ , according to [39]. Because $\theta(0_{X_{P}^{\ast},2})(z_{P}(dz_{P})^{1/2})=z_{P}^{2}(dz_{P})^{-1/2}\cdot(dz_{P}/z_{P})$ , we obtain $b_{1}(h)\geq b_{2}(h)$ . Because $\det(h)=1$ , we obtain $b_{1}(h)+b_{2}(h)+3=0$ . Moreover, according to the norm estimate of Simpson [39], if $h_{j}\in\mathop{\rm Harm}\nolimits(0_{X_{P}^{\ast},2})$ $(j=1,2)$ satisfy $b_{i}(h_{1})=b_{i}(h_{2})$ $(i=1,2)$ then $h_{1}$ and $h_{2}$ are mutually bounded.

For a hyperbolic metric $g_{P}$ of $X_{P}^{\ast}$ normalized as $\sqrt{-1}R(g_{P})+2\omega_{g_{P}}=0$ , we obtain $h^{(1)}(g_{P})\in\mathop{\rm Harm}\nolimits(0_{X_{P}^{\ast},2})$ induced by $g_{P}$ . For any $(b_{1},b_{2})\in{\mathbb{R}}^{2}$ such that $b_{1}\geq b_{2}$ and $b_{1}+b_{2}+3=0$ , there exists $a\in{\mathbb{R}}_{\geq 0}$ such that $b_{1}=\frac{a}{2}-\frac{3}{2}$ and $b_{2}=-\frac{a}{2}-\frac{3}{2}$ . Then, we can directly check that $b_{i}\bigl{(}h^{(1)}(g_{P,a})\bigr{)}=b_{i}$ . Then, we obtain the first claim. The second claim can be checked in the case of $g_{P,a}$ $(a\in{\mathbb{R}}_{\geq 0})$ . ∎

Take any ${\boldsymbol{a}}\in{\mathbb{R}}^{D}_{\geq 0}$ . We obtain $h_{P,a_{P}}=h^{(1)}(g_{P,a_{P}})\in\mathop{\rm Harm}\nolimits(0_{X_{P}^{\ast},2})$ corresponding to $g_{P,a_{P}}$ . Let $h_{X\setminus D}\in\mathop{\rm Harm}\nolimits(0_{X\setminus D,2})$ on $X\setminus D$ corresponding to the hyperbolic metric $g_{X\setminus D}$ . Let $X_{1,P}$ $(P\in D)$ be relatively compact neighbourhoods of $P$ in $X_{P}$ . There exists a $G_{2}$ -invariant Hermitian metric $h_{0}$ such that (i) $h_{0}=h_{P,a_{P}}$ on $X_{1,P}$ , (ii) $h_{0}=h_{X\setminus D}$ on $X\setminus\coprod_{P\in D}X_{P}$ , (iii) $\det(h_{0})=1$ . Note that the support of $F(h_{0})=R(h_{0})+[\theta(0_{X\setminus D,2}),\theta(0_{X\setminus D,2})^{\dagger}_{h_{0}}]$ is compact. We also note that $K^{-1/2}_{X\setminus D}$ is the unique proper Higgs subbundle of $\bigl{(}\mathbb{K}_{X\setminus D,2},\theta(0_{X\setminus D,2})\bigr{)}$ .

Let $h_{1}$ denote the Hermitian metric of $K^{-1/2}_{X\setminus D}$ induced by $h_{0}$ . Because $\mathop{\rm rank}\nolimits K^{-1/2}_{X\setminus D}=1$ , we obtain $F(h_{1})=R(h_{1})$ . Let us prove that

\int_{X\setminus D}\sqrt{-1}R(h_{1})=-\infty.

(135)

Let $h_{2}$ denote the Hermitian metric of $K^{-1/2}_{X\setminus D}$ induced by the hyperbolic metric $g_{X\setminus D}$ . By the assumption, we have

\int_{X\setminus D}\sqrt{-1}R(h_{2})=-2\int_{X\setminus D}\omega_{g_{X\setminus D}}=-\infty.

By the construction, $h_{2}=h_{1}$ holds on $X\setminus\coprod X_{P}$ . By Lemma 8.3, the volume of $\coprod_{P\in D}(X_{P}\setminus D)$ is finite with respect to $g_{X\setminus D}$ . We obtain

\int_{X\setminus\coprod X_{P}}\sqrt{-1}R(h_{1})=\int_{X\setminus\coprod X_{P}}\sqrt{-1}R(h_{2})=-\infty.

Because $h_{1|X_{1,P}}$ $(P\in D)$ are induced by $g_{P,a_{P}}$ , we obtain

\int_{X_{1,P}\setminus D}\sqrt{-1}R(h_{1})=\int_{X_{1,P}\setminus D}-2\omega_{g_{P,{\boldsymbol{a}}}}<0.

Because $\coprod_{P\in D}(X_{P}\setminus X_{1,P})$ is compact, we obtain

\sum_{P\in D}\int_{X_{P}\setminus X_{1,P}}\sqrt{-1}R(h_{1})<\infty.

In all, we obtain (135). As a result, $(\mathbb{K}_{X\setminus D,2},\theta(0_{X\setminus D,2}),h_{0})$ is analytically stable. By Proposition 2.11, there exists $h_{{\boldsymbol{a}}}\in\mathop{\rm Harm}\nolimits(0_{X\setminus D,2})$ such that $h_{{\boldsymbol{a}}}$ and $h_{0}$ are mutually bounded on $X\setminus D$ . The Kähler metric $g_{{\boldsymbol{a}}}$ on $K_{X\setminus D}^{-1}$ induced by $h_{{\boldsymbol{a}}}$ satisfies the desired conditions. ∎

Remark 8.4.

If there exists a complete Kähler metric $g_{X\setminus D}$ of $X\setminus D$ with finite volume, then there exist a compact Riemann surface $\overline{X}$ and an open embedding $X\setminus D\longrightarrow\overline{X}$ whose complement is a finite subset. (For example, see [11, 40], where much more general results are proved.) Therefore, we may apply the theory of tame harmonic bundles [39] due to Simpson to study a problem as in Proposition 8.1. The stability condition provides us with a constraint on ${\boldsymbol{a}}$ .

8.1.2. Rank $r$ case

Let $r$ be a positive integer larger than $2$ . Let $0_{X\setminus D,r}$ denote the $r$ -differential on $X\setminus D$ which is constantly $0$ . Let $h\in\mathop{\rm Harm}\nolimits(0_{X\setminus D,r})$ . Because $\theta(0_{X\setminus D,r})$ is nilpotent, the harmonic bundle $(\mathbb{K}_{X\setminus D,r},\theta(0_{X\setminus D,r}),h)$ is tame. As in the proof of Proposition 3.21, from $(\mathbb{K}_{X_{P}^{\ast},r},\theta(0_{X_{P}^{\ast},r}),h_{|X_{P}^{\ast}})$ , we obtain the locally free $\mathcal{O}_{X_{P}}$ -modules $\mathcal{P}^{h}_{a}\mathbb{K}_{X_{P}^{\ast},r}=\bigoplus_{i=1}^{r}\mathcal{P}_{a}^{h}K^{(r+1-2i)/2}_{X_{P}^{\ast}}$ $(a\in{\mathbb{R}})$ , and we can prove that there exists $b\in{\mathbb{R}}$ such that $z_{P}^{i}(dz_{P})^{(r+1-2i)/2}$ $(i=1,\ldots,r)$ are sections of $\mathcal{P}^{h}_{b}\mathbb{K}_{X_{P}^{\ast},r}$ . For $i=1,\ldots,r$ , we obtain the numbers

b_{P,i}(h):=\inf\bigl{\{}b\in{\mathbb{R}}\,\big{|}\,\mbox{$|z_{P}|^{b}\cdot|z_{P}^{i}(dz)_{P}^{(r+1-i)/2}|_{h}$ is bounded}\bigr{\}}.

Let ${\boldsymbol{b}}_{P}(h)$ denote the tuple $(b_{P,1}(h),\ldots,b_{P,r}(h))$ . Because $\theta(0_{X\setminus D,r})$ is nilpotent, we obtain $|\theta(0_{X\setminus D,r})|_{h}=O\bigl{(}(-\log|z_{P}|)^{-1}\bigr{)}dz_{P}/z_{P}$ with respect to $h$ . Because

\theta(0_{X_{P}^{\ast},r})(z_{P}^{i}(dz_{P})^{(r+1-2i)/2})=z_{P}^{i+1}(dz_{P})^{(r+1-2(i+1))/2}\cdot(dz_{P}/z_{P})

for $i=1,\ldots,r-1$ , we obtain $b_{P,i}(h)\geq b_{P,i+1}(h)$ $(i=1,\ldots,r-1)$ . Because $\det(h)=1$ , we obtain $\sum b_{P,i}(h)=-r(r+1)/2$ .

For simplicity, we assume that the Gaussian curvature of $g_{X\setminus D}$ is $-2$ . Note that $(\mathbb{K}_{X\setminus D,r},\theta(0_{X\setminus D,r}))$ is isomorphic to the $(r-1)$ -th symmetric product of $(\mathbb{K}_{X\setminus D,2},\theta(0_{X\setminus D,2}))$ . Let $h_{r}^{(1)}(g_{X\setminus D})\in\mathop{\rm Harm}\nolimits(0_{X\setminus D,r})$ denote the harmonic metric induced by $h^{(1)}(g_{X\setminus D})$ .

Proposition 8.5.

Suppose that ${\boldsymbol{b}}_{P}=(b_{P,1},\ldots,b_{P,r})\in{\mathbb{R}}^{r}$ $(P\in D)$ satisfy

b_{P,1}\geq\cdots\geq b_{P,r},\quad\sum b_{P,j}=-\frac{r(r+1)}{2}.

Then, there exists $h\in\mathop{\rm Harm}\nolimits(0_{X\setminus D,r})$ such that (i) ${\boldsymbol{b}}_{P}(h)={\boldsymbol{b}}_{P}$ $(P\in D)$ , (ii) For any neighbourhood $N$ of $D$ , $h$ and $h_{r}^{(1)}(g_{X\setminus D})$ are mutually bounded on $X\setminus N$ .

Proof Let us explain an outline of the proof.

Lemma 8.6.

There exists $h_{P,{\boldsymbol{b}}_{P}}\in\mathop{\rm Harm}\nolimits(0_{X_{P}^{\ast},r})$ such that ${\boldsymbol{b}}(h_{P,{\boldsymbol{b}}})={\boldsymbol{b}}_{P}$ .

Proof Let $Y$ be a compact Riemann surface whose genus $g(Y)$ is larger than $10\sum_{i}|b_{P,i}|+10r^{2}$ . Let $Q$ be a point of $Y$ . Let $(Y_{Q},z_{Q})$ be a holomorphic coordinate neighbourhood of $Q$ in $Y$ such that $z_{Q}(Q)=0$ . For any $a\in{\mathbb{R}}$ , $\mathcal{P}_{a}\bigl{(}\mathbb{K}_{Y_{Q},r}(\ast Q)\bigr{)}$ denote the locally free $\mathcal{O}_{Y_{Q}}$ -module obtained as follows:

\mathcal{P}_{a}\bigl{(}\mathbb{K}_{Y_{Q},r}(\ast Q)\bigr{)}=\bigoplus_{i=1}^{r}\mathcal{O}_{Y_{Q}}([a-b_{P,i}])z_{Q}^{i}(dz_{Q})^{(r+1-2i)/2}.

Here, for $c\in{\mathbb{R}}$ , $[c]$ denotes $\max\{n\in{\mathbb{Z}}\,|\,n\leq c\}$ . From $\mathcal{P}_{a}\bigl{(}\mathbb{K}_{Y_{Q},r}(\ast Q)\bigr{)}$ and $\mathbb{K}_{Y\setminus Q,r}$ , we obtain locally free $\mathcal{O}_{Y}$ -modules $\mathcal{P}_{a}\bigl{(}\mathbb{K}_{Y,r}(\ast Q)\bigr{)}$ . Thus, we obtain a filtered bundle $\mathcal{P}_{\ast}(\mathbb{K}_{Y,r}(\ast Q))$ on $(Y,Q)$ . We can easily check $\theta(0_{Y\setminus Q,r})\mathcal{P}_{a}(\mathbb{K}_{Y,r}(\ast Q))\subset\mathcal{P}_{a}(\mathbb{K}_{Y,r}(\ast Q))\otimes K_{Y}(Q)$ . We obtain

\deg\Bigl{(}\mathcal{P}_{\ast}\bigl{(}K^{(r+1-2i)/2}_{Y}(\ast Q)\bigr{)}\Bigr{)}=\frac{r+1-2i}{2}(2g(Y)-2)-(b_{P,i}+i).

For any $1\leq j\leq r$ , we obtain the following:

\sum_{i=j}^{r}\deg\Bigl{(}\mathcal{P}_{\ast}\bigl{(}K^{(r+1-2i)/2}_{Y}(\ast Q)\bigr{)}\Bigr{)}\\ =-\frac{(j-1)(r+1-j)}{2}(2g(Y)-2)-\sum_{i=j}^{r}(b_{P,i}+i).

(136)

Hence, for any $1<j\leq r$ , we obtain

\sum_{i=j}^{r}\deg\Bigl{(}\mathcal{P}_{\ast}\bigl{(}K^{(r+1-2i)/2}_{Y}(\ast Q)\bigr{)}\Bigr{)}<0=\deg\Bigl{(}\mathcal{P}_{\ast}\bigl{(}\mathbb{K}_{Y,r}(\ast Q)\bigr{)}\Bigr{)}.

If a non-zero subbundle $E\subset\mathbb{K}_{Y\setminus Q,r}$ satisfies $\theta(0_{Y\setminus Q,r})E\subset E\otimes K_{Y\setminus Q}$ , then there exists $1\leq j\leq r$ such that $E=\bigoplus_{i=j}^{r}K_{Y\setminus Q}^{(r+1-2i)/2}$ . Therefore, the regular filtered Higgs bundle $\bigl{(}\mathcal{P}_{\ast}\mathbb{K}_{Y,r}(\ast Q),\theta(0_{Y\setminus Q},r)\bigr{)}$ is stable. According to [39], there uniquely exists a harmonic metric $h_{Y,{\boldsymbol{b}}_{P}}$ of $(\mathbb{K}_{Y\setminus Q,r},\theta(0_{Y\setminus Q,r}))$ such that $\det(h_{Y,{\boldsymbol{b}}_{P}})=1$ and that $\mathbb{K}_{Y\setminus Q,r}$ with $h_{Y,{\boldsymbol{b}}_{P}}$ induces $\mathcal{P}_{\ast}\bigl{(}\mathbb{K}_{Y,r}(\ast Q)\bigr{)}$ . By the uniqueness, we obtain that $h_{Y,{\boldsymbol{b}}_{P}}$ is $G_{r}$ -invariant.

We embed $X_{P}$ into $Y$ by using $z_{P}$ and $(Y_{Q},z_{Q})$ . Then, we can construct $h_{P,{\boldsymbol{b}}_{P}}$ as the pull back of $h_{Y,{\boldsymbol{b}}_{P}}$ . ∎

Let $h_{0}$ be a $G_{r}$ -invariant Hermitian metric of $\mathbb{K}_{X\setminus D,r}$ satisfying the following conditions.

•

$h_{0}=h_{r}^{(1)}(g_{X\setminus D})$ on $X\setminus\coprod_{P\in D}X_{P}$ .
•

There exist relatively compact neighbourhoods $X_{1,P}$ $(P\in D)$ in $X_{P}$ such that $h_{0}=h_{P,{\boldsymbol{b}}_{P}}$ on $X_{1,P}\setminus\{P\}$ .
•

$\det(h_{0})=1$ .

Note that the support of $F(h_{0})$ is compact.

Lemma 8.7.

$(\mathbb{K}_{X\setminus D,r},\theta(0_{X\setminus D,r}),h_{0})$ is analytically stable with respect to the $G_{r}$ -action.

Proof For a $G_{r}$ -invariant Hermitian metric $h$ of $\mathbb{K}_{X\setminus D,r}$ , let $h_{\geq j}$ $(1\leq j\leq r)$ denote the induced Hermitian metric of $\bigoplus_{i=j}^{r}K^{(r+1-2i)/2}_{X\setminus D}$ . Because the volume of $X\setminus D$ with respect to $g_{X\setminus D}$ is infinite, we obtain the following for $1<j\leq r$ :

\int_{X\setminus D}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits F\bigl{(}h_{r}^{(1)}(g_{X\setminus D})_{\geq j}\bigr{)}=\\ \int_{X\setminus D}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits R(h_{r}^{(1)}\bigl{(}g_{X\setminus D})_{\geq j}\bigr{)}=-\infty.

(137)

By Lemma 8.3, and by $h_{0|X\setminus\coprod X_{P}}=h_{r}^{(1)}(g_{X\setminus D})_{|X\setminus\coprod X_{P}}$ , we obtain

\int_{X\setminus\coprod X_{P}}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits F\bigl{(}(h_{0})_{\geq j}\bigr{)}=\int_{X\setminus\coprod X_{P}}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits R\bigl{(}(h_{0})_{\geq j}\bigr{)}=-\infty.

(138)

On $X_{1,P}$ , $h_{0}=h_{P,{\boldsymbol{b}}_{P}}$ holds by the construction. Because $h_{P,{\boldsymbol{b}}_{P}}$ is harmonic, we obtain $F(h_{P,{\boldsymbol{b}}_{P}})=0$ . Let $\pi_{j}$ denote the projection of $\mathbb{K}_{X_{1,P},r}$ onto $\bigoplus_{i=j}^{r}K^{(r+1-2i)/2}_{X_{1,P}^{\ast}}$ . It is the orthogonal projection with respect to $h_{0}$ . By the Chern-Weil formula [38, Lemma 3.2], we obtain

\int_{X_{1,P}}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits F\bigl{(}(h_{0})_{\geq j}\bigr{)}=-\int_{X_{1,P}}\bigl{|}(\overline{\partial}_{E}+\theta)(\pi_{j})\bigr{|}^{2}_{h_{0}}\leq 0.

(139)

Because $X_{P}\setminus X_{1,P}$ is relatively compact, we obtain

\int_{X_{P}\setminus X_{1,P}}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits F\bigl{(}(h_{0})_{\geq j}\bigr{)}<\infty.

(140)

By (138), (139) and (140), we obtain

\int_{X\setminus D}\sqrt{-1}\Lambda\mathop{\rm Tr}\nolimits F\bigl{(}(h_{0})_{\geq j}\bigr{)}=-\infty<0.

If a non-zero subbundle $E$ of $\mathbb{K}_{X\setminus D,r}$ satisfies $\theta(0_{X\setminus D,r})E\subset E\otimes K_{X\setminus D}$ , there exists $1\leq j\leq r$ such that $E=\bigoplus_{i=j}^{r}K^{(r+1-2i)/2}_{X\setminus D}$ . Therefore, $(\mathbb{K}_{X\setminus D,r},\theta(0_{X\setminus D,r}),h_{0})$ is analytically stable with respect to the $G_{r}$ -action.

∎

By Proposition 2.11, there exists $h\in\mathop{\rm Harm}\nolimits(0_{X\setminus D,r})$ such that $h$ and $h_{0}$ are mutually bounded. Then, $h$ satisfies the desired conditions. ∎

8.2. Existence of harmonic metrics in the potential theoretically hyperbolic case

8.2.1. Solvability of the Poisson equation and the existence of harmonic metrics

Let $G$ be a compact Lie group with a character $\kappa:G\longrightarrow S^{1}$ . Let $X$ be an open Riemann surface equipped with a $G$ -action. Let $g_{X}$ be any $G$ -invariant Kähler metric of $X$ . Let $\Lambda$ denote the adjoint of the multiplication of the Kähler form associated with $g_{X}$ . Let $(E,\overline{\partial}_{E},\theta)$ be a $(G,\kappa)$ -homogeneous Higgs bundle on $X$ . Let $h_{0}$ be a $G$ -invariant Hermitian metric of $E$ .

A function on $X$ is called locally bounded if its restriction to any compact set is bounded.

Condition 8.8.

Assume that there exists a $G$ -invariant ${\mathbb{R}}_{\geq 0}$ -valued locally bounded function $\alpha$ on $X$ such that

\sqrt{-1}\Lambda\overline{\partial}\partial\alpha\geq\bigl{|}\Lambda F(h_{0})\bigr{|}_{h_{0}}

in the sense of distributions. Note that this condition is independent of the choice of $g_{X}$ .

Proposition 8.9.

There exists a $G$ -invariant harmonic metric $h$ of the Higgs bundle $(E,\overline{\partial}_{E},\theta)$ such that

\log\mathop{\rm Tr}\nolimits(h\cdot h_{0}^{-1})\leq\alpha+\log\mathop{\rm rank}\nolimits(E),\quad\log\mathop{\rm Tr}\nolimits(h_{0}\cdot h^{-1})\leq\alpha+\log\mathop{\rm rank}\nolimits(E).

(141)

Proof Let $X_{1}\subset X_{2}\subset\cdots$ be a smooth exhaustive sequence of $X$ . Let $h_{i}$ be a harmonic metric of $(E,\overline{\partial}_{E},\theta)_{|X_{i}}$ such that $h_{i|\partial X_{i}}=h_{0|\partial X_{i}}$ . Let $s_{i}$ be the automorphism of $E_{|X_{i}}$ determined by $h_{i}=h_{0|X_{i}}s_{i}$ . By (21), we obtain

\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm Tr}\nolimits(s_{i})\leq\Bigl{(}|F(h_{0})|_{h_{0}}\Bigr{)}_{|X_{i}}.

By the condition, we obtain the following on $X_{i}$ :

\sqrt{-1}\Lambda\overline{\partial}\partial\Bigl{(}\log\mathop{\rm Tr}\nolimits(s_{i})-\alpha\Bigr{)}\leq 0.

Note that $s_{i|\partial X_{i}}=\mathop{\rm id}\nolimits_{E_{i}|\partial X_{i}}$ . Hence, we obtain

\log\mathop{\rm Tr}\nolimits(s_{i})\leq\alpha_{|X_{i}}+\log(\mathop{\rm rank}\nolimits E)-\min_{\partial X_{i}}(\alpha)\leq\alpha_{|X_{i}}+\log(\mathop{\rm rank}\nolimits E).

Similarly, we obtain

\log\mathop{\rm Tr}\nolimits(s_{i}^{-1})\leq\alpha_{|X_{i}}+\log(\mathop{\rm rank}\nolimits E).

Then, we obtain the claim of the proposition. ∎

Corollary 8.10.

If $\det(h_{0})$ is flat, there exists a $G$ -invariant harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)$ satisfying (141) and $\det(h)=\det(h_{0})$ . Moreover, if $\alpha$ is bounded, and if $\int_{X}|F(h_{0})|<\infty$ , then $(\overline{\partial}_{E}+\theta)(h\cdot h_{0}^{-1})$ is $L^{2}$ . ∎

Remark 8.11.

In [34], the solvability of the Poisson equation is applied to the solvability of the Hermitian-Einstein equation. Proposition 8.9 is an analogue in the context of harmonic metrics on Riemann surfaces.

8.2.2. Potential theoretically hyperbolic Riemann surfaces

Let $X$ be a potential theoretically hyperbolic Riemann surface, i.e., there exists a non-constant non-positive subharmonic function on $X$ . It is well known that there exists a Green function ${\texttt{G}}(p,q):X\longrightarrow{\mathbb{R}}_{<0}$ characterized as follows (see [43, §7]):

•

${\texttt{G}}(p,\bullet)$ is harmonic on $X\setminus\{p\}$ .
•

Let $(X_{p},z_{p})$ be a holomorphic coordinate neighbourhood such that $z_{p}(p)=0$ . Then, ${\texttt{G}}(p,\bullet)-\log|z_{p}|^{2}$ is harmonic on $X_{p}$ .
•

If $H$ is any other subharmonic function satisfying the above two conditions, then ${\texttt{G}}(p,\bullet)\geq H$ .

Recall that for any compact support $2$ -form $\psi$ on $X$ we obtain

\sqrt{-1}\partial\overline{\partial}\Bigl{(}\frac{1}{2\pi}\int{\texttt{G}}(p,q)\psi(q)\Bigr{)}=\psi.

Lemma 8.12.

Let $g$ be any Kähler metric of $X$ . The volume form is denoted by $\mathop{\rm dvol}\nolimits$ . For a function $v:X\longrightarrow{\mathbb{R}}_{\geq 0}$ , suppose that $|{\texttt{G}}(p,q)(v\cdot\mathop{\rm dvol}\nolimits)(q)|$ is integrable on $X$ for any $p$ . Then, by setting $\alpha(v)(p):=-\frac{1}{2\pi}\int{\texttt{G}}(p,q)(v\mathop{\rm dvol}\nolimits)(q)$ , we obtain $\alpha(v)\geq 0$ and $\sqrt{-1}\Lambda\overline{\partial}\partial\alpha(v)=v$ . ∎

The following is clear by the construction of the Green function (see [43, §7]).

Lemma 8.13.

Let $K$ be any compact subset of $X$ . Let $N$ be any relatively compact neighbourhood of $K$ in $X$ . Then, ${\texttt{G}}(p,q)$ is bounded on $K\times(X\setminus N)$ . ∎

Corollary 8.14.

Let $\psi$ be a compact support $2$ -form on $X$ . Then, there exists a bounded function $\alpha$ on $X$ such that $\partial\overline{\partial}\alpha=\psi$ . ∎

Let $(E,\overline{\partial}_{E},\theta)$ be a $(G,\kappa)$ -homogeneous Higgs bundle on $X$ . Let $h_{0}$ be a $G$ -invariant Hermitian metric of $E$ . Suppose that

\int{\texttt{G}}(p,q)\bigl{(}|F(h_{0})|_{h_{0}}\bigr{)}\mathop{\rm dvol}\nolimits(q)<\infty

for any $p$ .

Proposition 8.15.

There exists a $G$ -invariant harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)$ such that

\max\Bigl{\{}\log\bigl{|}h\cdot h_{0}^{-1}\bigr{|}_{h_{0}},\,\,\log\bigl{|}h_{0}\cdot h^{-1}\bigr{|}_{h_{0}}\Bigr{\}}\leq\alpha(|F(h_{0})|_{h_{0}})+\log\mathop{\rm rank}\nolimits(E).

If $\det(h_{0})$ is flat, $h$ also satisfies $\det(h)=\det(h_{0})$ . ∎

Corollary 8.16.

Suppose that the support of $F(h_{0})$ is compact. Then, there exists a $G$ -invariant harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)$ such that $h$ and $h_{0}$ are mutually bounded. If $\det(h_{0})$ is flat, it also satisfies (i) $\det(h)=\det(h_{0})$ , (ii) $(\overline{\partial}_{E}+\theta)(h\cdot h_{0}^{-1})$ is $L^{2}$ . ∎

8.2.3. Upper half plane

Let us state a consequence in the case $X=\mathbb{H}=\{z\in{\mathbb{C}}\,|\,\mathop{\rm Im}\nolimits(z)>0\}$ . Let $(x,y)$ be the real coordinate system determined by $z=x+\sqrt{-1}y$ .

Proposition 8.17.

Let $h_{0}$ be a $G$ -invariant Hermitian metric of $E$ such that $|F(h_{0})|=O((1+y)^{-2})$ . Then, there exists a $G$ -invariant harmonic metric $h$ of $(E,\overline{\partial}_{E},\theta)$ such that

\max\Bigl{\{}\log\bigl{|}h\cdot h_{0}^{-1}\bigr{|}_{h_{0}},\,\log\bigl{|}h_{0}\cdot h^{-1}\bigr{|}_{h_{0}}\Bigr{\}}=O\Bigl{(}\log(2+y)\Bigr{)}.

If $\det(h_{0})$ is flat, we obtain $\det(h)=\det(h_{0})$ . If $|F(h_{0})|=O((1+y)^{-2-\epsilon})$ for a positive constant $\epsilon$ , then $h$ and $h_{0}$ are mutually bounded.

Proof For a function $f$ on $\mathbb{H}$ such that $f=O\bigl{(}(1+y)^{-2}\bigr{)}$ , we set

u(x,y)=-\frac{1}{4\pi}\int_{\mathbb{H}}\log\Bigl{(}\frac{(x-\xi)^{2}+(y-\eta)^{2}}{(x-\xi)^{2}+(y+\eta)^{2}}\Bigr{)}f(\xi,\eta)\,d\xi\,d\eta.

Then, $-(\partial_{x}^{2}+\partial_{y}^{2})u=f$ . If $f$ is positive, then $u$ is also positive on $\mathbb{H}$ . The rest follows from Lemma 4.5 and Proposition 8.9. ∎

8.3. Uniqueness of harmonic metrics in the potential theoretically parabolic case

Let $X$ be a potential theoretically parabolic Riemann surface, i.e., $X$ is non-compact, and any non-positive subharmonic function on $X$ is constant. Let $(E,\overline{\partial}_{E},\theta)$ be a Higgs bundle on $X$ .

Proposition 8.18.

Let $h_{i}$ $(i=1,2)$ be harmonic metrics of the Higgs bundle $(E,\overline{\partial}_{E},\theta)$ which are mutually bounded. Then, there exists a decomposition

(E,\overline{\partial}_{E},\theta)=\bigoplus_{j=1}^{m}(E_{j},\overline{\partial}_{E_{j}},\theta_{j})

such that (i) the decomposition is orthogonal with respect to both $h_{1}$ and $h_{2}$ , (ii) there exist positive numbers $a_{j}$ such that $h_{1|E_{j}}=a_{j}h_{2|E_{j}}$ .

Proof Let $s$ be the automorphism of $E$ determined by $h_{2}=h_{1}s$ . Then, $\log\mathop{\rm Tr}\nolimits(s)$ is a bounded subharmonic function on $X$ . Because $X$ is assumed to be potential theoretically parabolic, $\log\mathop{\rm Tr}\nolimits(s)$ is constant. According to (19), it implies that $(\overline{\partial}_{E}+\theta)s=0$ , and hence we obtain the desired decomposition. ∎

The following corollary of the proposition has been well known after [38, 39].

Corollary 8.19.

Let $X$ be the complement of a finite subset $D$ of a compact Riemann surface $\overline{X}$ . Then, the claim of Proposition 8.18 holds.

Proof Let $f$ be an ${\mathbb{R}}_{\leq 0}$ -valued subharmonic function on $X$ . According to [36, Theorem 3.3.25], it uniquely extends to a subharmonic function on $\overline{X}$ . By the maximum principle, we obtain that $f$ is constant, i.e., $X$ is potential theoretically parabolic. ∎

Remark 8.20.

It is also easy and standard to prove the claim of the corollary directly. Indeed, in the proof of Proposition 8.18, without any assumption on $X$ , we obtain the boundedness of $\log\mathop{\rm Tr}\nolimits(s)$ and $\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm Tr}\nolimits(s)\leq 0$ on $X$ . If $X=\overline{X}\setminus D$ , $\sqrt{-1}\Lambda\overline{\partial}\partial\log\mathop{\rm Tr}\nolimits(s)\leq 0$ holds on $\overline{X}$ according to [39, Lemma 2.2]. Hence, $\log\mathop{\rm Tr}\nolimits(s)$ is constant, and we obtain $(\overline{\partial}_{E}+\theta)s=0$ .

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Isolated singularities of Toda equations and cyclic Higgs bundles

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

1.1. Harmonic bundles and Toda equations

1.1.1. Higgs bundles associated with rr-differentials and harmonic metrics

1.1.2. Toda equations associated with rr-differentials

Remark 1.1.

Remark 1.2.

Remark 1.3.

1.1.3. Existence and uniqueness of complete solutions

Theorem 1.4 ([24]).

1.1.4. Uniqueness and non-uniqueness of general solutions

Proposition 1.5 ([24]).

Remark 1.6.

Proposition 1.7 ([24]).

1.1.5. Remark on the difference of the conventions for the induced Hermitian metrics

1.2. Isolated singularities

1.2.1. The case of poles

1.2.2. Holomorphic functions with multiple growth orders

Remark 1.8.

1.2.3. The case where qq is not meromorphic but has multiple growth orders

Theorem 1.9 (Theorem 6.1).

1.3. Global classification in the case where qq has multiple growth orders

Theorem 1.10.

Corollary 1.11.

Theorem 1.12 (Theorem 6.6).

1.4. Comparison of the methods and the results

1.5. A generalization

Remark 1.13.

Proposition 1.14 (Proposition 7.5, Proposition 7.6).

Theorem 1.15 (Theorem 7.11).

1.6. Examples on ℂ{\mathbb{C}}

Proposition 1.16 (A special case of Proposition 6.12).

Proposition 1.17 (A special case of Proposition 6.14).

Corollary 1.18.

Proposition 1.19 (Proposition 6.16).

1.7. Acknowledgement

2. Some existence results of harmonic metrics

2.1. Dirichlet problem

Proposition 2.1 (Donaldson).

Corollary 2.2.

2.1.1. Homogeneous case

Lemma 2.3.

2.1.2. Donaldson functional

Lemma 2.4.

2.2. Convergence

Definition 2.5.

Proposition 2.6.

Proposition 2.7.

Remark 2.8.

2.3. A Kobayashi-Hitchin correspondence

Condition 2.9.

Definition 2.10.

Proposition 2.11.

Lemma 2.12.

2.4. Appendix

3. Preliminaries for harmonic metrics of cyclic Higgs bundles

3.1. Cyclic Higgs bundles associated with rr-differentials

Remark 3.1.

3.2. Hermitian metrics on a vector space with a cyclic automorphism

Lemma 3.2.

3.2.1. Canonical metric

Lemma 3.3.

3.2.2. ϵ\epsilon-orthogonality

3.2.3. Norm of the automorphism

Lemma 3.4.

Corollary 3.5.

Corollary 3.6.

3.3. Harmonic metrics of a cyclic Higgs bundle on a disc

Proposition 3.7 ([32, Proposition 2.1]).

Corollary 3.8.

Proposition 3.9 ([32, Corollary 2.6]).

Corollary 3.10.

3.3.1. Estimates near boundary

Proposition 3.11.

Proposition 3.12.

3.3.2. Appendix: A variant of Simpson’s main estimate near boundary

Proposition 3.13.

1.1.1. Higgs bundles associated with $r$ -differentials and harmonic metrics

1.1.2. Toda equations associated with $r$ -differentials

1.2.3. The case where $q$ is not meromorphic but has multiple growth orders

1.3. Global classification in the case where $q$ has multiple growth orders

1.6. Examples on ${\mathbb{C}}$

3.1. Cyclic Higgs bundles associated with $r$ -differentials

3.2.2. $\epsilon$ -orthogonality

3.5.1. The case $m\leq 0$

3.5.2. The case $m>0$

3.6. Examples on ${\mathbb{C}}$