Gluing Harmonic Maps

Loosely speaking, we are given two compact Riemannian manifolds of dimension 2, $\Sigma_{1}$ and $\Sigma_{2}$, a Riemannian manifold $N$, and two harmonic maps $f_{1}:\Sigma_{1}\rightarrow N$ and $f_{2}:\Sigma_{2}\rightarrow N$. Suppose $f_{1}(x_{1})=f_{2}(x_{2})$, then we connect $\Sigma_{1}$ and $\Sigma_{2}$ by punching holes at $x_{1}$ and $x_{2}$, then gluing them by a neck. Denote this by $\Sigma_{1}\#_{\delta,R}\Sigma_{2}$, where $\delta$ and $R$ are parameters of neck. We glue the two maps using cutoff functions.

Here are the details: Consider gluing $f^{0}$ and $f^{\infty}$ whose domains are both $S^{2}$. We denote

$$y:=f^{0}(0)=f^{\infty}(\infty)$$

Since $f^{0}$ and $f^{\infty}$ are harmonic maps, we know they are smooth. Since $S^{2}$ is compact, we know there exists $c>0$ such that $\|df^{0}\|_{L^{\infty}}\leq c$ and $\|df^{\infty}\|_{L^{\infty}}\leq c$ (Here the norms are in the sense of the round metric).

Let $\epsilon$ be less than the injective radius of $N$. By the upper bound of the differential of the harmonic maps, we can compute that when $\cot\frac{\epsilon}{c}<|z|<\tan\frac{\epsilon}{c}$, we have $d(f^{0}(z),y)<\epsilon$ and $d(f^{\infty}(z),y)<\epsilon$. Thus there exists $\zeta^{0}(z),\zeta^{\infty}(z)\in T_{y}N$ such that $f^{0}(z)=\exp_{y}(\zeta^{0}(z))$ and $f^{\infty}(z)=\exp_{y}(\zeta^{\infty}(z))$.

Consider some fixed nondecreasing smooth function $\rho$ satisfying:

$$\rho(z)=\left\{\begin{aligned} &0,|z|\leq 1\\ &1,|z|\geq 2\\ \end{aligned}\right.$$

We define our pre-gluing:

(1)

$$f^{R}(z):=f^{\delta,R}(z)=\left\{\begin{aligned} &f^{0}(z),&|z|\geq\frac{2}{\delta R}\\ &\exp_{y}\left(\rho(\delta Rz)\zeta^{0}(z)+\rho\left(\frac{\delta}{Rz}\right)\zeta^{\infty}(R^{2}z)\right),&\frac{\delta}{2R}\leq|z|\leq\frac{2}{\delta R}\\ &f^{\infty}(R^{2}z),&|z|\leq\frac{\delta}{2R}\end{aligned}\right.$$

We will need to specify which weighted norm we are using. For the reason mentioned in section 10.3 in [16], we use the same weight as in chapter 10 in [16]. Namely,

$$\theta^{R}(z)=\left\{\begin{aligned} &R^{-2}+R^{2}|z|^{2}&|z|\leq\frac{1}{R}\\ &1+|z|^{2}\ &|z|\geq\frac{1}{R}\\ \end{aligned}\right.$$

and we let the metric be

(2)

$$g=(\theta^{R})^{-2}(ds^{2}+dt^{2})$$

Note that similar setup can be done for general closed Riemannian manifolds of dimension 2.

Given Riemannian manifolds $\Sigma$ and $N$, where $\Sigma$ is closed and of dimension 2. Consider a smooth map $f:\Sigma\rightarrow N$. For $\xi\in f^{-1}TN$, we can consider the perturbation of $f$ by $\xi$ under the exponential map, which we write as $\exp_{f}(\xi)$. We define $W^{2,p}_{f}$ to be the space consisting all $\xi$ that is $W^{2,p}$ in each coordinate chart, with the usual Sobolev space structure. We can define $L^{p}$ spaces similarly.

We define the harmonic map operator $\mathcal{F}$ from the space $W^{2,p}_{f}$ to $L^{p}_{f}$ as

$$\mathcal{F}_{f}(\xi):=\Phi_{f}(\xi)^{-1}\left((\triangle_{g}(\exp_{f}(\xi))^{k}+g^{\alpha\beta}(\Gamma^{N})^{k}_{ij}(\exp_{f}(\xi))(\exp_{f}(\xi))^{i}_{\alpha}(\exp_{f}(\xi))^{j}_{\beta})\frac{\partial}{\partial y^{k}}\right)\\$$

where $g$ denotes the metric on $\Sigma$, and $\Phi$ is the parallel transport from $\exp_{f}(\xi)$ to $f$ along the geodesic $\exp_{f}(t\xi)$. See Section 2.1 for details. We will deal with a second order nonlinear system of PDEs.

First, we define the space we will be working with, which is a space of pairs of harmonic maps satisfying certain conditions. These conditions will allow us to carry out a construction similar to that in chapter 10 of [16].

In this paper, we will need to specify the parameters of the neck in connected sum. To avoid repeating too much, we define the following symbol.

We will show that for pairs of harmonic maps in definition 1.1, we can glue the pair and then perturb it to get another harmonic map. This is stated as the following theoerm, which is proved in Section 6:

Consider bubbling described in [21]. By derivative estimates and facts about the bubbling phenomenon, we will eventually show the following using the result in Section 8:

In Section 2, we introduce the setup of the problem. We define the harmonic map operator and compute the linearization. Then we explore the structure of the space of harmonic maps, which behaves locally like a smooth manifold. In Section 3, we introduce the implicit function theorem needed in the gluing construction. In Section 4, we explain how to choose the appropriate metric and the cutoff functions to glue the domain manifolds. In Section 5, we construct an approximate right inverse, and then get the exact right inverse using Newton iteration. In Section 6, we apply the implicit function theorem to find a way to perturb the preglued map into a harmonic map. In Section 7, we cite a way to ’suspend’ the harmonic map into a conformal harmonic map, which will be used in the proof of surjectivity. In Section 8, by derivative estimates and properties of conformal harmonic maps, we show that the gluing map constructed in Section 6 is locally surjective under certain conditions, and finally we will be able to apply this to the situation of bubbling to show that, under certain conditions, the bubble lies in the image of the gluing map. The appendix contains technical details including proof of Lemma 2.1, norm and derivative estimates, as well as related facts in elliptic PDE theory and functional analysis.

The framework for Sections 4, 5, and the first part of 8 is similar to chapter 10 of [16]. However, we are dealing with a second order nonlinear system of PDEs on Riemannian manifolds while in [16] they have a first order PDE on symplectic manifolds.

The author expresses deep gratitude to Professor Paul Feehan for his invaluable guidance and unwavering support, and to Professor Dan Ketover for his insightful comments and suggestions. Special thanks are also extended to Professor Jason Lotay, Zilu Ma, Gregory Parker, Junsheng Zhang, Xiao Ma, Liuwei Gong, and Jiakai Li for their helpful discussions. This work is also based in part on research supported by the National Science Foundation under Grant No. 1440140, while the author was in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during Fall 2022 as an associate of the program Analytic and Geometric Aspects of Gauge Theory.

Let’s follow chapter 10 in [16] and consider gluing harmonic maps on $S^{2}$ (The Riemann sphere) or any Riemann manifolds of dimension 2. For simplicity, we consider the case when $\Sigma_{1}=S^{2},\Sigma_{2}=S^{2}$. However, we will see that the gluing can be used for general Riemann manifolds of dimension 2.

First, let’s consider the general definition of harmonic maps. Suppose $M$, $N$ are Riemannian manifolds and $f$ is a smooth map from the domain manifold $M$ to the target manifold $N$. We say $f$ is a harmonic map if it satisfies the harmonic map equation. There are multiple ways of writing the harmonic map equation. One would be

(3)

$$P(f):=\triangle_{g}f+A(f)(df,df)$$

where we consider an isometric embedding $N\subset\mathbb{R}^{N}$ and $A$ is the second fundamental from of the embedding.

However, the above will depend on the ambient Euclidean space. In particular, since we are using the implicit function theorem later, the above will make it difficult to utilize the surjectivity onto the tangent space. An alternate form would be

$$tr_{g}(\nabla df)=0$$

In coordinates, this is

$$g^{\alpha\beta}\left(f^{k}_{\alpha\beta}-(\Gamma^{M})^{\gamma}_{\alpha\beta}f^{k}_{\gamma}+(\Gamma^{N})^{k}_{ij}(f)f^{i}_{\alpha}f^{j}_{\beta}\right)\frac{\partial}{\partial y^{k}}=0$$

where we are denoting $\frac{\partial f}{\partial x^{\alpha}}$ by $f_{\alpha}$. We know that the above is a well defined vector, independent of choice of coordinates on $M$ and $N$. We can also verify this by directly computing the transformation law in different coordinates.

In this paper we use another form obtained from considering coordinate charts in $M$ and computing the Euler-Lagrange equation of the energy. See section 1.1 from [13] for details. The equation is

$$\triangle_{g}f^{k}+g^{\alpha\beta}(\Gamma^{N})^{k}_{ij}(f)f^{i}_{\alpha}f^{j}_{\beta}=0$$

or, equivalently,

$$g^{\alpha\beta}\frac{\partial^{2}f^{k}}{\partial x^{\alpha}\partial x^{\beta}}+\frac{\partial g^{\alpha\beta}}{\partial x^{\alpha}}\frac{\partial f^{k}}{\partial x^{\beta}}+g^{\alpha\beta}\frac{1}{2\det g}\frac{\partial\det g}{\partial x^{\alpha}}\frac{\partial f^{k}}{\partial x^{\beta}}+g^{\alpha\beta}(\Gamma^{N})^{k}_{ij}(f)f^{i}_{\alpha}f^{j}_{\beta}=0$$

We define the operator $P$ as

$$P(f):=(\triangle_{g}f^{k}+g^{\alpha\beta}(\Gamma^{N})^{k}_{ij}(f)f^{i}_{\alpha}f^{j}_{\beta})\frac{\partial}{\partial y^{k}}$$

It can be verified directly by coordinate change that the above is a tensor.

Next, we consider the linearization of the above differential operator. The idea is that, for a perturbation of $f$ and $x\in M$, the operator will spit out a vector that belongs to a different fiber in the tangent bundle. Thus we have to pull back the vector before comparing them.

Here are the details: Given $\xi\in W^{2,p}(\Sigma,f^{-1}TN)$, we know that for $1<p<2$, $W^{2,p}$ is embedded into $C^{0,\gamma}$ where $\gamma=2-\frac{2}{p}$. We define the following:

(4)			$\displaystyle\Phi_{f}(\xi):f^{-1}TN\rightarrow(\exp_{f}(\xi))^{-1}TN$
(5)			$\displaystyle\mathcal{F}_{f}(\xi):=\Phi_{f}(\xi)^{-1}\left((\triangle_{g}(\exp_{f}(\xi))^{k}+g^{\alpha\beta}(\Gamma^{N})^{k}_{ij}(\exp_{f}(\xi))(\exp_{f}(\xi))^{i}_{\alpha}(\exp_{f}(\xi))^{j}_{\beta})\frac{\partial}{\partial y^{k}}\right)$
(6)			$\displaystyle D_{f}:=d\mathcal{F}_{f}(0)$

We would like to compute $D_{f}$. Consider the path:

		$\displaystyle\mathbb{R}\rightarrow W^{2,p}(\Sigma,N)$
		$\displaystyle\lambda\mapsto\exp_{f}(\lambda\xi)$
		$\displaystyle\Phi_{f}(\lambda\xi)\mathcal{F}_{f}(\lambda\xi)=(\triangle_{g}(\exp_{f}(\xi))^{k}+g^{\alpha\beta}(\Gamma^{N})^{k}_{ij}(\exp_{f}(\xi))(\exp_{f}(\xi))^{i}_{\alpha}(\exp_{f}(\xi))^{j}_{\beta})\frac{\partial}{\partial y^{k}}$
	$\displaystyle D_{f}\xi$	$\displaystyle=\frac{d}{d\lambda}\mathcal{F}_{f}(\lambda\xi)\mid_{\lambda=0}$
		$\displaystyle=\nabla_{\lambda}P(\exp_{f}(\lambda\xi))$

We use normal coordinates for $f(x)\in N$ to get an explicit expression for $D_{f}$:

Consider $\{\frac{\partial}{\partial y^{i}}\}$ a basis of $TN$ in normal coordinates. Using $f$ we can view it as a basis of $f^{-1}TN$. $P(f)\in\Gamma(f^{-1}TN)$ is locally given by

$$(\triangle_{g}f^{i}+g^{\alpha\beta}\Gamma_{jl}^{i}(f(x))\frac{\partial f^{l}}{\partial x^{\alpha}}\frac{\partial f^{j}}{\partial x^{\beta}})\frac{\partial}{\partial y^{i}}$$

where $\Gamma$ is the Christoffel symbol on $N$.

In normal coordinates, $\exp_{f}(\xi)$ is just $f+\xi$. We have

$$P(\exp_{f}(\lambda\xi))=(\triangle_{g}(f^{i}+\lambda\xi^{i})+g^{\alpha\beta}\Gamma_{jl}^{i}(f(x)+\lambda\xi)\frac{\partial(f^{j}+\lambda\xi^{j})}{\partial x^{\alpha}}\frac{\partial(f^{l}+\lambda\xi^{l})}{\partial x^{\beta}})\frac{\partial}{\partial y^{i}}$$

We denote $(\triangle_{g}(f^{i}+\lambda\xi^{i})+g^{\alpha\beta}\Gamma_{jl}^{i}(f(x)+\lambda\xi)\frac{\partial(f^{j}+\lambda\xi^{j})}{\partial x^{\alpha}}\frac{\partial(f^{l}+\lambda\xi^{l})}{\partial x^{\beta}})$ by $\phi_{i}$ and $\frac{\partial}{\partial y^{i}}$ by $e_{i}$. Since we are using normal coordinates, we have

	$\displaystyle\nabla_{\xi}(\phi_{i}e_{i})$	$\displaystyle=(\nabla_{\xi}\phi_{i})\cdot e_{i}$
		$\displaystyle=\left(\triangle_{g}\xi^{i}+g^{\alpha\beta}\partial_{s}\Gamma_{jl}^{i}\xi^{s}\frac{\partial f^{j}}{\partial x^{\alpha}}\frac{\partial f^{l}}{\partial x^{\beta}}\right)e_{i}$

Thus we get

(7)

$$D_{f}\xi=\left(\triangle_{g}\xi^{i}+g^{\alpha\beta}\partial_{s}\Gamma_{jl}^{i}(f(x))\xi^{s}\frac{\partial f^{j}}{\partial x^{\alpha}}\frac{\partial f^{l}}{\partial x^{\beta}}\right)e_{i}$$

Note that in the above we are only able to compute the exact form of the operators when considering the normal coordinates at exactly that point. For certain estimates, we will need to consider the operator’s coordinate form in a single coordinate chart. When we change the coordinates, it will be different for every point in the neighborhood, so we will need some uniform bound on the change of coordinates.

The proof is in the appendix.

Using (7) and lemma 2.1, we get (note that above we are considering a different coordinate system at every point, but below we locally use the same coordinate chart):

(8)

$\displaystyle D_{f}\xi=\left(\triangle_{g}\xi^{k}+F_{1}^{k}\left(f,\frac{\partial f}{\partial x},\frac{\partial^{2}f}{\partial x^{2}},\xi\right)+F_{2}^{k}\left(f,\frac{\partial f}{\partial x},\frac{\partial\xi}{\partial x}\right)\right)e_{k}$

where

	$\displaystyle F_{1}^{k}\left(f,\frac{\partial f}{\partial x},\frac{\partial^{2}f}{\partial x^{2}},\xi\right)=$	$\displaystyle g^{\alpha\beta}\left(\frac{\partial^{3}\tilde{y}^{i}}{\partial y^{s}\partial y^{l}\partial y^{m}}\frac{\partial f^{l}}{\partial x^{\alpha}}\frac{\partial f^{m}}{\partial x^{\beta}}+\frac{\partial^{2}\tilde{y}^{i}}{\partial y^{l}\partial y^{s}}\frac{\partial^{2}f^{l}}{\partial x^{\alpha}\partial x^{\beta}}\right)\frac{\partial y^{k}}{\partial\tilde{y}^{i}}\xi^{s}+$
		$\displaystyle\frac{\partial g^{\alpha\beta}}{\partial x^{\alpha}}\frac{\partial^{2}\tilde{y}^{i}}{\partial y^{l}\partial y^{s}}\frac{\partial f^{s}}{\partial x^{\beta}}\frac{\partial y^{k}}{\partial\tilde{y}^{i}}\xi^{l}+g^{\alpha\beta}\frac{\partial\det g}{\partial x^{\alpha}}\frac{1}{2\det g}\frac{\partial^{2}\tilde{y}^{i}}{\partial y^{l}\partial y^{s}}\frac{\partial f^{s}}{\partial x^{\beta}}\frac{\partial y^{k}}{\partial\tilde{y}^{i}}\xi^{l}+$
		$\displaystyle g^{\alpha\beta}\xi^{r}\frac{\partial}{\partial y^{r}}\left(\frac{\partial\tilde{y}^{i}}{\partial y^{\eta}}\frac{\partial y^{\theta}}{\partial\tilde{y}^{j}}\frac{\partial y^{\phi}}{\partial\tilde{y}^{l}}\Gamma^{\eta}_{\theta\phi}+\frac{\partial^{2}y^{\eta}}{\partial\tilde{y}^{j}\partial\tilde{y}^{l}}\frac{\partial\tilde{y}^{i}}{\partial y^{\eta}}\right)\frac{\partial\tilde{y}^{j}}{\partial y^{m}}\frac{\partial f^{m}}{\partial x^{\alpha}}\frac{\partial\tilde{y}^{l}}{\partial y^{s}}\frac{\partial f^{s}}{\partial x^{\beta}}$

and

$$F_{2}^{k}\left(f,\frac{\partial f}{\partial x},\frac{\partial\xi}{\partial x}\right)=g^{\alpha\beta}\frac{\partial f^{l}}{\partial x^{\alpha}}\frac{\partial\xi^{s}}{\partial x^{\beta}}\frac{\partial^{2}\tilde{y}^{i}}{\partial y^{l}\partial y^{s}}\frac{\partial y^{k}}{\partial\tilde{y}^{i}}$$

Note that by lemma 2.1, the coordinate change functions are uniformly bounded.

This part follows from section 4.2.4 of [5] or section A.4 in [16]. Suppose $D_{f^{0}}:W^{2,p}(S^{2},(f^{0})^{-1}TN)\rightarrow L^{p}(S^{2},(f^{0})^{-1}TN)$ is Fredholm. It follows that the kernel and image of $D_{f^{0}}$ are closed and admit topological complements. For simplicity, we write $U:=W^{2,p}(S^{2},(f^{0})^{-1}TN)$ and $V:=L^{p}(S^{2},(f^{0})^{-1}TN)$. So we can write $U=U_{0}\oplus F$, $V=V_{0}\oplus G$, where $F$ and $G$ are finite-dimensional and $D_{f^{0}}$ is a linear isomorphism from $U_{0}$ to $V_{0}$.

Let $N$ be a connected open neighburhood of $0$ in $U$. By definition we know $\mathcal{F}$ is Fredholm. If $D_{f^{0}}$ is surjective, then by the implicit function theorem of Banach spaces we know there is a diffeomorphism $\psi$ from one neighborhood of $0$ in $U$ to another, such that $\mathcal{F}\circ\psi=D_{f^{0}}$. Furthermore, we know that

Now consider the general case when $D_{f^{0}}$ is not necessarily surjective. Consider projection of $V$ onto $V_{0}$, we will have the derivative be surjective. Then we obtain

As an immediate corollary we obtain a finite-dimensional model for a neighborhood of $0$ in the zero set $Z(\mathcal{F})$. Note that elements in $Z(\mathcal{F})$ are smooth by elliptic regularity (use partition of unity to reduce the case on manifolds to that on Euclidean space).

Explicitly, there exists a $C^{\infty}$ diffeomorphism $g$ from some open set $W$ containing $0\in W^{2,p}(S^{2},(f^{0})^{-1}TN)$, such that

$$g(0)=0,\quad dg(0)=id$$

and

$$\mathcal{F}^{-1}(0)\cap g(W)=g(W\cap\ker D)$$

From the above we can get a coordinate chart for $g(W)$ by the isomorphism between $\ker D$ and $\mathbb{R}^{m}$, where $m$ is the dimension of the kernel. Thus we can view the vectors in the kernel as tangent vectors of the moduli space. Furthermore, a smooth path $f_{t}$ in the open neighborhood can be represented as $g(\gamma_{1}(t)\xi_{1}+\cdots+\gamma_{m}(t)\xi_{m})$. Similarly, for a sufficiently small open neighborhood, we can find a smooth frame.

In the setting for theorem 1.1, since the linearizations are assumed to be surjective, together with proposition C.2 in the appendix, we know that the above can be applied.

Consider $\mathcal{M}(c,p)$, note that for linear operators between Banach spaces, surjective operators form open sets in the sense of operator norm. Thus locally we can still find charts as above.