Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

Longzhi Lin and Jingyong Zhu Longzhi Lin: Mathematics Department, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA [email protected] Jingyong Zhu: Department of Mathematics, Sichuan University, No.24 South Section 1, Yihuan Road, Chengdu, 610065, China [email protected]

Abstract.

Motivated by the theory of harmonic maps on Riemannian surfaces, conformal-harmonic maps between two Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ were introduced in search of a natural notion of “harmonicity” for maps defined on a general even dimensional Riemannian manifold $\mathcal{M}$ . They are critical points of a conformally invariant energy functional and reassemble the GJMS operators when the target is the set of real or complex numbers. On a four dimensional manifold, conformal-harmonic maps are the conformally invariant counterparts of the intrinsic bi-harmonic maps and a mapping version of the conformally invariant Paneitz operator for functions.

In this paper, we consider conformal-harmonic maps from locally conformally flat 4-manifolds into spheres. We prove a quantitative uniqueness result for such conformal-harmonic maps as an immediate consequence of convexity for the conformally-invariant energy functional. To this end, we are led to prove a version of second order Hardy inequality on manifolds, which may be of independent interest.

Key words and phrases:

conformal-harmonic map, uniqueness, Hardy inequality

2020 Mathematics Subject Classification:

58E15, 35J58, 58E20

1. Introduction

The most prominent and classic problem in calculus of variations for mappings between two Riemannian manifolds $(\mathcal{M},g)$ and $(\mathcal{N},h)\hookrightarrow\mathbf{R}^{K}$ is the study of harmonic maps, which are critical points $u:\mathcal{M}\to\mathcal{N}$ of the Dirichlet energy

E_{1}(u,\mathcal{M}):=\int_{\mathcal{M}}|du|^{2}\omega_{g}\,,

where $\omega_{g}$ is the volume measure on $\mathcal{M}$ defined by the metric $g$ and $|du|^{2}$ is the Hilbert-Schmidt norm square of $du$ . The conformal invariance of $E_{1}(u,\mathcal{M})$ on a Riemannian surface $\mathcal{M}$ (with respect to the metric $g$ of $\mathcal{M}$ ) and its connection to the theory of minimal surfaces make harmonic maps on two dimensional domains the most widely studied topic in the field of geometric analysis ever since the pioneering work of J. Eells and J. Sampson [13], see also [19]. Motivated by the theory of harmonic maps on Riemannian surfaces, the Paneitz operator [39] and GJMS operators [17], in a recent work [2] V. Bérard has shown the existence of an intrinsically defined energy functional for smooth maps between two Riemannian manifolds $(\mathcal{M}^{2m},g)$ (of even dimension $2m$ where $m\geq 1$ ) and $\mathcal{N}$ , denoted by $E_{m}(u,\mathcal{M})$ , which is conformally invariant with respect to $g$ and coincides with the above Dirichlet energy $E_{1}(u,\mathcal{M})$ when $m=1$ . Following the terminology of A. Gastel and A. Nerf in [15] who considered an extrinsic analogue (i.e., a variant dependent of the embedding $\mathcal{N}\hookrightarrow\mathbf{R}^{K}$ ) of $E_{m}(u,\mathcal{M})$ , we will call $E_{m}(u,\mathcal{M})$ an intrinsic Paneitz energy when $m=2$ and intrinsic Paneitz poly-energy when $m\geq 3$ . The critical points of $E_{m}(u,\mathcal{M})$ are called conformal-harmonic maps or C-harmonic maps, which generalize the harmonic maps on surfaces (i.e., $m=1$ ) and satisfy a system of nonlinear PDEs with leading term $\Delta^{m}$ . When $\mathcal{N}=\mathbf{R}$ or $\mathbf{C}$ , the induced operators for the critical points reassemble the GJMS operators [17], see also Chang-Yang [6]. In particular, when $m=2$ the intrinsic Paneitz energy reads as

(1.1)

\mathcal{E}(u,\mathcal{M}):=E_{2}(u,\mathcal{M}^{4})=\int_{\mathcal{M}}|\tau(u)|^{2}+\frac{2}{3}{\rm Sc}^{\mathcal{M}}|du|^{2}-2{\rm Ric}^{\mathcal{M}}(du,du)\,,

where $\tau(u)\in u^{*}T\mathcal{N}$ is the tension field, we denote the scalar curvature and Ricci curvature of $\mathcal{M}$ by ${\rm Sc}^{\mathcal{M}}$ and ${\rm Ric}^{\mathcal{M}}$ , respectively. We remark that $\mathcal{E}(u,\mathcal{M})$ is conformally invariant on a four dimensional manifold $\mathcal{M}$ (see Appendix A) and critical points of $\mathcal{E}(u,\mathcal{M})$ are the conformal-invariant counterparts of the intrinsic bi-harmonic maps, i.e., the critical points of the intrinsic bi-energy

\mathcal{F}(u,\mathcal{M}):=\int_{\mathcal{M}}|\tau(u)|^{2}\,.

Conformal-harmonic maps are also generalizations of the Paneitz operator in the context of maps, and intrinsically they satisfy the following system of $4$ -th order PDEs (c.f. [22, 27, 37]):

\Delta^{u}\tau(u)=R^{\mathcal{N}}(\nabla u,\tau(u))\nabla u+\frac{2}{3}\left({\rm Sc}^{\mathcal{M}}\tau(u)+\nabla{\rm Sc}^{\mathcal{M}}\nabla u\right)-2{\rm div}({\rm Ric}^{\mathcal{M}}(\nabla u,\cdot)),

where $\Delta^{u}$ is the induced Laplace operator on the pullback vector bundle $u^{\ast}T\mathcal{N}$ over $\mathcal{M}$ or equivalently and extrinsically (c.f. [32, equation (1.2)], [31, equation (9)])

	$\displaystyle\Delta^{2}u=$	$\displaystyle-\Delta(\nabla P^{\bot}\nabla u)-{\rm div}(\nabla P^{\bot}\Delta u)+2\nabla P^{\bot}\nabla(\nabla P^{\bot}\nabla u)$
		$\displaystyle+2\nabla P^{\bot}\nabla P^{\bot}\Delta u-(\nabla PP^{\bot}-P^{\bot}\nabla P)\nabla\Delta u$
(1.2)			$\displaystyle+P\langle\nabla P^{\bot}\nabla u,{D}_{u}({D}_{u}P^{\bot})\nabla u\nabla u\rangle-2{\rm div}\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}P\rangle$
		$\displaystyle+2\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}\nabla P\rangle+\frac{2}{3}{\rm Sc}^{\mathcal{M}}(\Delta u+\nabla P^{\perp}\nabla u)$
		$\displaystyle+\frac{2}{3}\nabla{\rm Sc}^{\mathcal{M}}\nabla u-2\nabla\text{Ric}^{\mathcal{M}}(\nabla u,\cdot)-2{\rm Ric}^{\mathcal{M}}(P(\nabla^{2}u),\cdot)\,,$

see Appendix B for a detailed derivation of this equation. Here $R^{\mathcal{N}}$ is the curvature tensor of $\mathcal{N}$ , $\nabla$ and $\Delta$ are the Levi-Civita connection and Laplace–Beltrami operator on $\mathcal{M}$ respectively, $\tau(u)=(\Delta u)^{T}$ is the tension field, $P(u)$ is the orthogonal projection to the tangent plane $T_{u}\mathcal{N}$ , $D$ is the derivative with respect to the standard coordinates of $\mathbf{R}^{K}$ and $A(\cdot,\cdot)$ denotes the second fundamental form of $\mathcal{N}\hookrightarrow\mathbf{R}^{K}$ . Note that this conformal-harmonic map equation (1) differs from the intrinsic biharmonic map equation by lower order terms which make (1) conformally invariant.

Higher order geometric variational problems, including the study of (both extrinsic and intrinsic) biharmonic maps and polyharmonic maps, have attracted much attention in the last two decades, see e.g. [11, 5, 43, 45, 29, 40, 42, 16, 30, 15, 35, 1, 18, 10] for the extrinsic case and [45, 25, 38, 16, 32] for the intrinsic case. The corresponding heat flows have been studied extensively as well, as a tool to prove the existence of biharmonic maps and polyharmonic maps in a given homotopy class, see e.g. [26, 27, 14, 46, 20, 21, 32]. It should be noted that the extrinsic and intrinsic cases come in two different flavors: the intrinsic variants are considered more geometrically natural because they do not depend on the embedding of the target manifold $\mathcal{N}\hookrightarrow\mathbf{R}^{K}$ , although they are less natural from the variational point of view due to the lack of coercivity for the intrinsic energies (and thus they are considerably more difficult analytically and less studied); on the other hand, the extrinsic variants are more natural from the analytical point of view but in turn they do depend on the embedding of $\mathcal{N}\hookrightarrow\mathbf{R}^{K}$ . Among them, the conformally invariant problems (both extrinsic and intrinsic) are considered the most geometric, see e.g., [11, 15]. Most of the results in the current literature concern the regularity and existence of (the weak solutions of) the critical points of the higher order geometric variational problems because they are associated to systems of higher order PDEs with critical growth nonlinearities. However, the uniqueness problem of these critical points have been left largely open. In [32] P. Laurain and the first author proved a version of bienergy convexity (and thus the uniqueness) for weakly intrinsic biharmonic maps in $W^{2,2}(B_{1},\mathbf{S}^{n})$ with small bienergy and prescribed boundary data, where $B_{1}\subset\mathbf{R}^{4}$ is the unit 4-ball and $\mathbf{S}^{n}\subset\mathbf{R}^{n+1}$ is the standard unit sphere. We shall remark that such energy convexity plays an essential role in the discrete replacement min-max construction of geometric objects of interest (such as minimal spheres and free boundary minimal disks in manifolds), see e.g. Colding-Minicozzi [8], Lamm-Lin [28], Lin-Sun-Zhou [34] and Laurain-Petrides [33].

We shall remark that the intrinsic Paneitz energy functional $\mathcal{E}(u,\mathcal{M})$ defined in (1.1) on $\mathbf{S}^{4}$ was already used by T. Lamm in [27] as a tool to prove that every weakly intrinsic biharmonic maps from $\mathbf{R}^{4}$ into a non-positively curved target manifold with finite bienergy has to be constant. More recently, O. Biquard and F. Madani in [3] used the corresponding heat flow for $\mathcal{E}(u,\mathcal{M})$ to prove an existence result for conformal-harmonic maps from a certain class of 4-manifolds into a non-positively curved manifold $\mathcal{N}$ . Here the conformal-harmonic map heat flow (or, the negative $L^{2}$ -gradient flow of $\mathcal{E}(\cdot,\mathcal{M})$ ) is defined as follow (when $\partial\mathcal{M}=\emptyset$ ):

(1.3)

\left\{\begin{aligned} \frac{\partial u}{\partial t}+\Delta^{u}\tau(u)=&R^{\mathcal{N}}(\nabla u,\tau(u))\nabla u+\frac{2}{3}\left({\rm Sc}^{\mathcal{M}}\tau(u)+\nabla{\rm Sc}^{\mathcal{M}}\nabla u\right)\\ &-2{\rm div}({\rm Ric}^{\mathcal{M}}(\nabla u,\cdot))\quad\text{ on }\mathcal{M}\times[0,T)\\ u=&\,u_{0}\quad\text{ on }\mathcal{M}\times\{0\}\,,\end{aligned}\right.

and (when $\partial\mathcal{M}\neq\emptyset$ )

(1.4)

\left\{\begin{aligned} \frac{\partial u}{\partial t}+\Delta^{u}\tau(u)=&R^{\mathcal{N}}(\nabla u,\tau(u))\nabla u+\frac{2}{3}\left({\rm Sc}^{\mathcal{M}}\tau(u)+\nabla{\rm Sc}^{\mathcal{M}}\nabla u\right)\\ &-2{\rm div}({\rm Ric}^{\mathcal{M}}(\nabla u,\cdot))\quad\text{ on }\mathcal{M}\times[0,T)\\ u=&\,u_{0}\quad\text{ on }\mathcal{M}\times\{0\}\\ u=&\,u_{0}\,,\quad\partial_{\nu}u=\partial_{\nu}u_{0}\quad\text{ on }\partial\mathcal{M}\times[0,T)\,,\end{aligned}\right.

where $u_{0}\in C^{\infty}\cap W^{2,2}(\overline{\mathcal{M}},\mathcal{N})$ . More precisely, in [3] O. Biquard and F. Madani proved

Theorem 1.1.

Let $(\mathcal{M},g)$ and $(\mathcal{N},h)$ be compact Riemannian manifolds of four dimensions and $n$ dimensions respectively. Assume that $\mathcal{N}$ has non-positive curvature, the Yamabe invariant

\mu(\mathcal{M},[g]):=\inf_{g^{\prime}\in[g]}\frac{\int_{\mathcal{M}}\text{Sc}_{g^{\prime}}dv_{g^{\prime}}}{\left(\int_{\mathcal{M}}dv_{g^{\prime}}\right)^{1/2}}>0

and the conformally invariant total $Q$ -curvature

\kappa(\mathcal{M},[g]):=\frac{1}{12}\int_{\mathcal{M}}\left(\text{Sc}^{2}_{g}-3|\text{Ric}_{g}|^{2}\right)dv_{g}

satisfies $\kappa+\frac{1}{6}\mu^{2}>0$ . Then the conformal-harmonic map heat flow 1.3 exists and is smooth for all time and converges subsequently to a smooth conformal-harmonic map $u_{\infty}(\mathcal{M},\mathcal{N})$ as $t\to\infty$ . Consequently, there exists a conformal-harmonic map in each homotopy class in $C^{\infty}(\mathcal{M},\mathcal{N})$ .

In this paper, we consider the intrinsic Paneitz energy $\mathcal{E}(u,\mathcal{M})$ for $W^{2,2}$ maps from locally conformally flat 4-manifolds with smooth boundaries into spheres and study the (quantitative) uniqueness of the critical points. The first main result of this paper is a version of energy convexity for $\mathcal{E}(u,\mathcal{M})$ , more precisely, we prove

Theorem 1.2.

Let $(\mathcal{M},g)$ be a four dimensional compact locally conformally flat Riemannian manifold with a smooth boundary $\partial\mathcal{M}$ . There exist $\epsilon_{0}>0,C>0$ depending only on $\mathcal{M}$ such that for any $u,v\in W^{2,2}(\mathcal{M},\mathbf{S}^{n})$ with $u=v$ , $\partial_{\nu}u=\partial_{\nu}v$ on $\partial\mathcal{M}$ ,

(1.5)

\int_{\mathcal{M}}|\Delta u|^{2}dv_{g}\leq\epsilon_{0},\quad\int_{\mathcal{M}}|\nabla v|^{4}dv_{g}\leq\epsilon_{0}

and $u$ is a weakly conformal-harmonic map, we have

(1.6)

\frac{1}{C}\int_{\mathcal{M}}|\Delta v-\Delta u|^{2}dv_{g}\leq\mathcal{E}(v,\mathcal{M})-\mathcal{E}(u,\mathcal{M})\,.

Remark 1.3.

Our proof relies locally on the result of Laurain and the first author [32], which was established on the unit ball in Euclidean space. Our results could also extend to general source and target manifolds if the Laurain-Lin result is generalized to such settings.

In order to prove this theorem, we are led to show a second order Hardy inequality on smooth manifolds with smooth boundaries. Inspired by the recent work of D’Ambrosio-Dipierro [9] (which followed the techniques introduced by Mitidieri in [36]), we are able to prove the following version of second order Hardy inequality on certain smooth manifolds, for details see Section 3.

Theorem 1.4.

Let $(\mathcal{M},g)$ be an $n$ dimensional compact Riemannian manifold with a smooth boundary $\partial\mathcal{M}$ and $\rho\in W^{1,2}_{loc}(\mathcal{M},\mathbf{R})$ be a nonnegative function such that $\Delta\rho\leq 0$ on $\mathcal{M}$ in the weak sense and $|\nabla\rho|\geq\delta$ a.e. on $\mathcal{M}$ for some $\delta>0$ , then there exits $C:=C(\mathcal{M})>0$ such that

\int_{\mathcal{M}}\frac{|w|^{2}|\nabla\rho|^{2}}{\rho^{4}}dv_{g}\leq C(1+\delta^{-2})^{2}\int_{\mathcal{M}}|\Delta w|^{2}dv_{g}

for any $w\in C^{\infty}_{0}(\mathcal{M},\mathbf{R})$ .

In fact, based on Theorem 1.4 we are able to prove a general version of second order Hardy inequality that is valid on any smooth manifold.

Theorem 1.5.

Let $(\mathcal{M},g)$ be an $n$ dimensional compact Riemannian manifold with a smooth boundary $\partial\mathcal{M}$ and $\rho$ be the distance function to the boundary. Then there exits $C:=C(\mathcal{M})>0$ such that

\int_{\mathcal{M}}\frac{|w|^{2}}{\rho^{4}}dv_{g}\leq C\int_{\mathcal{M}}|\Delta w|^{2}dv_{g}

for any $w\in C^{\infty}_{0}(\mathcal{M},\mathbf{R})$ .

Proof.

Let $\psi>0$ be the eigenfunction associated to the first eigenvalue $\lambda_{1}>0$ of $-\Delta_{g}$ on $\mathcal{M}$ with $\psi|_{\partial\mathcal{M}}=0$ . Then we have $\psi\in C^{\infty}(\overline{\mathcal{M}})$ and $|\nabla\psi|>0$ on $\partial{\mathcal{M}}$ by the Hopf boundary point Lemma (see e.g. [7, Theorem E.4]), noting that $\frac{\partial\psi}{\partial\nu}<0$ and the tangential derivative of $\psi$ vanishes on the boundary. Now as in the proof of [9, Theorem 6.2], for fixed $\gamma>0$ , define

\mathcal{M}_{\gamma}=\{x\in\mathcal{M}:\rho(x)\geq\gamma\}

and

\mathcal{M}^{\gamma}=\{x\in\mathcal{M}:\rho(x)<\gamma\}.

Then there exist constants $\epsilon,L,\delta>0$ depending on $\mathcal{M}$ and a smooth cut-off function $\chi\in C^{\infty}(\overline{\mathcal{M}},\mathbf{R})$ such that

\psi(x)\leq L\rho(x),\quad x\in\mathcal{M}\quad\text{and}\quad|\nabla\psi|(x)\geq\delta,\quad x\in\mathcal{M}^{2\epsilon}\,,

$\chi=1$ on $\overline{\mathcal{M}^{\epsilon}}$ , $\chi=0$ on $M\setminus\mathcal{M}^{2\epsilon}$ , and $|\nabla^{i}\chi|\leq C\epsilon^{-i},i=1,2$ on $\mathcal{M}^{2\epsilon}\setminus\mathcal{M}^{\epsilon}$ . For any $w\in C^{\infty}_{0}(\mathcal{M},\mathbf{R})$ , denote

\tilde{w}=w\cdot\chi\in C^{\infty}_{0}(\mathcal{M}^{2\epsilon},\mathbf{R}).

Then by Theorem 1.4 (replacing $\mathcal{M}$ by $\mathcal{M}^{2\epsilon}$ , $w$ by $\tilde{w}$ and $\rho$ by $\psi$ ) we get

	$\displaystyle\int_{\mathcal{M^{\epsilon}}}\frac{\|w\|^{2}}{\rho^{4}}dv_{g}$	$\displaystyle\leq\frac{L^{4}}{\delta^{2}}\int_{\mathcal{M}^{2\epsilon}}\frac{\|\tilde{w}\|^{2}\|\nabla\psi\|^{2}}{\psi^{4}}dv_{g}$
		$\displaystyle\leq\frac{C(\mathcal{M})(1+\delta^{-2})^{2}L^{4}}{\delta^{2}}\int_{\mathcal{M}^{2\epsilon}}\|\Delta\tilde{w}\|^{2}dv_{g}$
(1.7)			$\displaystyle\leq\frac{C(\mathcal{M},\epsilon)(1+\delta^{-2})^{2}L^{4}}{\delta^{2}}\int_{\mathcal{M}}\|\Delta w\|^{2}dv_{g}\,.$

Now let $m_{\epsilon}=\min_{\mathcal{M}_{\epsilon}}\psi>0,D=\max_{\mathcal{M}}\psi>0$ , then by [9, Theorem 6.3] (with $p=2,s=\frac{1}{2}$ ) we have

	$\displaystyle\int_{\mathcal{M_{\epsilon}}}\frac{\|w\|^{2}}{\rho^{4}}dv_{g}$	$\displaystyle\leq m_{\epsilon}^{-\frac{1}{2}}\epsilon^{-4}\int_{\mathcal{M}}\|w\|^{2}\psi^{\frac{1}{2}}dv_{g}$
		$\displaystyle\leq 8m_{\epsilon}^{-\frac{1}{2}}\epsilon^{-4}\lambda_{1}^{-1}D^{\frac{1}{2}}\int_{\mathcal{M}}\|\nabla w\|^{2}dv_{g}$
(1.8)			$\displaystyle\leq C(\mathcal{M},\epsilon,m_{\epsilon},\lambda_{1},D)\int_{\mathcal{M}}\|\Delta w\|^{2}dv_{g}\,.$

Combining (1) and (1) completes the proof. ∎

As an immediate corollary of Theorem 1.2 we get the following uniqueness result for weakly conformal-harmonic maps from a locally conformally flat $4$ -manifold $\mathcal{M}$ into spheres.

Corollary 1.6.

Let $(\mathcal{M},g)$ be a four dimensional compact locally conformally flat Riemannian manifold with a smooth boundary $\partial\mathcal{M}$ . Then there exists $\epsilon_{0}>0$ depending only on $\mathcal{M}$ such that for any weakly conformal-harmonic maps $u,v\in W^{2,2}(\mathcal{M},\mathbf{S}^{n})$ with $u=v$ , $\partial_{\nu}u=\partial_{\nu}v$ on $\partial\mathcal{M}$ and

\int_{\mathcal{M}}|\Delta u|^{2}dv_{g}\leq\epsilon_{0},\quad\int_{\mathcal{M}}|\Delta v|^{2}dv_{g}\leq\epsilon_{0},

we have $u\equiv v$ on $\mathcal{M}$ .

Acknowledgement

We would like to thank the referees for their valuable comments, which have significantly improved the presentation of the results in this paper. The first author acknowledges partial support from a COR research funding at UC Santa Cruz. The second author would like to thank the support by the National Natural Science Foundation of China (Grant No. 12201440) and the Fundamental Research Funds for the Central Universities (Grant No. YJ2021136).

2. Preliminary

In this section, we will fix some notations and recall a technical theorem ( $\epsilon$ -regularity) that will be used later. Throughout this section, $(\mathcal{M},g)$ denotes a smooth $4$ -dimensional compact Riemannian manifold with a smooth boundary $\partial\mathcal{M}$ and $(\mathcal{N},h)$ is an $n$ -dimensional smooth closed Riemannian manifold which can be embedded into $\mathbf{R}^{K}$ . As mentioned in the introduction, a weakly conformal-harmonic map $u$ from $\mathcal{M}$ into $\mathcal{N}\hookrightarrow{\mathbf{R}}^{k}$ is a map in $W^{2,2}(\mathcal{M},{\mathbf{R}}^{k})$ that is a critical point of the conformally invariant energy $\mathcal{E}(\cdot,\mathcal{M})$ defined in (1.1) and takes values almost everywhere in $\mathcal{N}$ .

Note that the dimension $4$ is critical for the analysis of weakly conformal-harmonic maps (e.g. a $W^{2,2}$ map falls in $L^{p}$ for any $p<\infty$ but barely fails to be continuous in dimension $4$ ). Now let $\Pi:\mathcal{N}_{\delta}\rightarrow\mathcal{N}$ be the nearest point projection map, which is well defined and smooth for $\delta>0$ small enough. Here $\mathcal{N}_{\delta}=\{y\in{\mathbf{R}}^{k}\,|\,\text{dist}\,(y,\mathcal{N})\leq\delta\}$ . For $y\in\mathcal{N}$ , let

P(y)\equiv D\Pi(y):{\mathbf{R}}^{k}\rightarrow T_{y}\mathcal{N}

be the orthogonal projection onto the tangent plane $T_{y}\mathcal{N}$ , and

P^{\bot}(y)\equiv\text{Id}-D\Pi(y):{\mathbf{R}}^{k}\rightarrow(T_{y}\mathcal{N})^{\bot}\,,

where $D$ is the derivative with respect to the standard coordinates of $\mathbf{R}^{K}$ . In the following, we will write $P$ (resp. $P^{\bot}$ ) instead of $P(y)$ (resp. $P^{\bot}(y)$ ) and we will identify these linear transformations with their matrix representations in $M_{n}$ . We also note that these projections are in $W^{2,2}(\mathcal{M},M_{n})$ as soon as $u$ is in $W^{2,2}(\mathcal{M},\mathcal{N})$ . Finally, note that the second fundamental form $A(\,\cdot\,)(\,\cdot\,,\,\cdot\,)$ of $\mathcal{N}\subset{\mathbf{R}}^{k}$ is defined by

A(y)(Y,Z):=D_{Y}P^{\bot}(y)(Z),\quad\forall\,y\in\mathcal{N}\quad\text{and}\quad Y,Z\in T_{y}\mathcal{N}.

We know that $u=(u^{1},...,u^{k})\in W^{2,2}(\mathcal{M},\mathcal{N})$ is an intrinsic bi-harmonic map if it satisfies the fourth order PDE (see [44] and [31] for details)

	$\displaystyle\Delta^{2}u=$	$\displaystyle-\Delta(\nabla P^{\bot}\nabla u)-{\rm div}(\nabla P^{\bot}\Delta u)+2\nabla P^{\bot}\nabla(\nabla P^{\bot}\nabla u)$
		$\displaystyle+2\nabla P^{\bot}\nabla P^{\bot}\Delta u-(\nabla PP^{\bot}-P^{\bot}\nabla P)\nabla\Delta u$
(2.1)			$\displaystyle+P\langle\nabla P^{\bot}\nabla u,{D}_{u}({D}_{u}P^{\bot})\nabla u\nabla u\rangle-2{\rm div}\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}P\rangle$
		$\displaystyle+2\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}\nabla P\rangle\,.$

Here $D_{u}P^{\bot}=D_{y}P^{\bot}(y)|_{y=u}$ . Note that

(2.2)

\nabla P^{\perp}=D_{u}P^{\perp}\nabla u\,,

and the following two terms in (2) are equivalent to:

(2.3)

\nabla P^{\perp}P={D}_{u}P^{\perp}\nabla uP=A(u)(\nabla u,P)

and

(2.4)

\nabla P^{\perp}\nabla P={D}_{u}P^{\perp}\nabla u\nabla P=A(u)(\nabla u,\nabla P).

When $\mathcal{N}=\mathbf{S}^{n}$ , we have (note that $\Delta u=\tau(u)-A(u)(\nabla u,\nabla u)$ ).

(2.5)

P^{\perp}(\Delta u)=-\nabla P^{\perp}\nabla u=-u|\nabla u|^{2}\,,

and therefore

(2.6)

A(u)(\nabla u,\nabla u)=\nabla P^{\perp}\nabla u=u|\nabla u|^{2}.

In particular, when $\mathcal{N}=\mathbf{S}^{n}$ , the intrinsic bi-harmonic map equation can be rewritten as (see e.g. Lamm-Rivière [29])

(2.7)

\Delta^{2}u\,=\,\Delta(V\cdot\nabla u)+\text{div }(w\nabla u)+W\cdot\nabla u\,,

where

(2.8)

\left\{\begin{aligned} V^{ij}&=u^{i}\nabla u^{j}-u^{j}\nabla u^{i}\\ w^{ij}&=\text{div}\left(V^{ij}\right)\\ W^{ij}&=\nabla w^{ij}+2\left[\Delta u^{i}\nabla u^{j}-\Delta u^{j}\nabla u^{i}+|\nabla u|^{2}(u^{i}\nabla u^{j}-u^{j}\nabla u^{i})\right]\,.\end{aligned}\right.

Remark 2.1.

In local coordinates, the terms in (2) read as

	$\displaystyle P\left(\langle\nabla P^{\bot}\nabla u,D_{u}(D_{u}P^{\bot})\nabla u\nabla u\rangle\right)$
	$\displaystyle=\sum_{\alpha,\beta,\gamma,i,j,k,m}P_{lk}\nabla_{\alpha}(P^{\bot})_{ij}\nabla_{\alpha}u^{j}D_{u^{k}}D_{u^{\beta}}(P^{\bot})_{im}\nabla_{\gamma}u^{\beta}\nabla_{\gamma}u^{m};$

	$\displaystyle\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}\nabla P\rangle\,$	$\displaystyle=\,\sum_{\alpha,\beta,i,j,k}\nabla_{\alpha}(P^{\bot})_{ij}\nabla_{\alpha}u^{j}\nabla_{\beta}(P^{\bot})_{ik}\nabla_{\beta}P_{kl}\,;$
	$\displaystyle\text{div}\,\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}P\rangle\,$	$\displaystyle=\,\sum_{\alpha,\beta,i,j,k}\nabla_{\beta}\left[\nabla_{\alpha}(P^{\bot})_{ij}\nabla_{\alpha}u^{j}\nabla_{\beta}(P^{\bot})_{ik}P_{kl}\right]\,,$

where $\nabla_{\alpha}:=\nabla_{e_{\alpha}}$ and $\{e_{\alpha}\}_{\alpha=1,\cdots,4}$ is a local orthonormal frame on $\mathcal{M}$ .

To end this section, let us recall the following version of $\varepsilon$ -regularity for approximate intrinsic and extrinsic bi-harmonic maps into spheres, which will be useful later. Throughout the rest of this paper, $\bar{\nabla}$ and $\bar{\Delta}$ will denote the connection and Laplacian with respect to the Euclidean metric.

Theorem 2.2.

([32, Theorem A.4], c.f. [29, 31]) Let $B_{1}\subset\mathbf{R}^{4}$ be the unit 4-ball. There exist ${\varepsilon}>0$ , $0<\delta<1$ , $\alpha>0$ and $C>0$ independent of $u$ such that if $u\in W^{2,2}(B_{1},\mathbf{S}^{n})$ is a solution of

(2.9)

\bar{\Delta}^{2}u=\bar{\Delta}(V\bar{\nabla}u)+\text{div}(w\bar{\nabla}u)+\bar{\nabla}\omega\bar{\nabla}u+F\bar{\nabla}u,

where $V\in W^{1,2}(B_{1},M_{n+1}\otimes\Lambda^{1}{\mathbf{R}}^{4})$ , $w\in L^{2}(B_{1},M_{n+1})$ , $\omega\in L^{2}(B_{1},\mathrm{s}o_{n+1})$ and $F\in L^{2}\cdot W^{1,2}(B_{1},M_{n+1}\otimes\Lambda^{1}{\mathbf{R}}^{4})$ , which satisfy

(2.10)

\begin{split}|V|&\leq C|\bar{\nabla}u|\,,\\ |F|&\leq C|\bar{\nabla}u|\left(|\bar{\nabla}^{2}u|+|\bar{\nabla}u|^{2}\right)\,,\\ |w|+|\omega|&\leq C\left(|\bar{\nabla}^{2}u|+|\bar{\nabla}u|^{2}\right)\end{split}

almost everywhere (where $C>0$ is a constant depending only on $\mathcal{N}$ ) and

(2.11)

\|\bar{\Delta}u\|_{L^{2}(B_{1})}\leq{\varepsilon},

then we have $u\in W_{loc}^{3,4/3}(B_{1},{\mathbf{R}}^{n+1})$ and

\|\bar{\nabla}^{3}u\|_{L^{\frac{4}{3}}(B(p,\rho))}+\|\bar{\nabla}^{2}u\|_{L^{2}(B(p,\rho))}+\|\bar{\nabla}u\|_{L^{4}(B(p,\rho))}\leq C\rho^{\alpha}\|\bar{\Delta}u\|_{L^{2}(B_{1})}

for all $p\in B_{\frac{1}{4}}$ and $0\leq\rho\leq\delta$ . Moreover, $u\in W^{3,\infty}(B_{\frac{1}{16}},{\mathbf{R}}^{n+1})$ and for $l=1,2,3$ we have

(2.12)

|\bar{\nabla}^{l}u|(0)\leq C_{l}\|\bar{\Delta}u\|_{L^{2}(B_{1})}

for some constant $C_{l}>0$ . In particular, by rescaling we have for $x\in B_{1}$ and $l=1,2,3$ :

(2.13)

|\bar{\nabla}^{l}u|(x)\leq C_{l}(1-|x|)^{-l}\|\bar{\Delta}u\|_{L^{2}(B_{1})}\,.

3. Second-order Hardy inequality on manifolds

The first order Hardy inequality on Euclidean domains is well-known. Let’s recall the second order Hardy inequality on Euclidean domains (see e.g. [12, 23]).

Theorem 3.1.

([12, Theorem 2]) Let $\Omega\subset\mathbf{R}^{n}$ be a bounded Lipschitz domain. There exists a constant $C>0$ depending only on $\Omega$ such that if $w\in W^{2,2}_{0}(\Omega,\mathbf{R}^{K})$ , then we have

(3.1)

\int_{\Omega}|w|^{2}(\text{dist}(x,\partial\Omega))^{-4}dx\leq C\int_{\Omega}|\bar{\Delta}w|^{2}dx.

Recently, in [9] D’Ambrosio and Dipierro extended the first order Hardy inequality to the Riemannian manifold setting (c.f. [4, 24]), more precisely, among other things they proved:

Theorem 3.2.

[9, Theorem 2.1] Let $(\mathcal{M},g)$ be an $n$ dimensional compact Riemannian manifold with a smooth boundary $\partial\mathcal{M}$ . Let $\rho\in W^{1,2}_{loc}(\mathcal{M},\mathbf{R})$ be a nonnegative function such that $\Delta\rho\leq 0$ on $\mathcal{M}$ in the weak sense, then we have $\frac{|\nabla\rho|^{2}}{|\rho|^{2}}\in L^{1}_{loc}(\mathcal{M},\mathbf{R})$ and

\int_{\mathcal{M}}\frac{|w|^{2}}{\rho^{2}}|\nabla\rho|^{2}dv_{g}\leq 4\int_{\mathcal{M}}|\nabla w|^{2}dv_{g}

for any $w\in C^{\infty}_{0}(\mathcal{M},\mathbf{R})$ .

Building on the work of D’Ambrosio and Dipierro [9], which utilizes techniques introduced by Mitidieri in [36], we establish a general version of the second-order Hardy inequality applicable to any smooth manifold (Theorem 1.5). Since Theorem 1.5 relies on Theorem 1.4, which is valid for certain smooth manifolds, we first provide a proof of Theorem 1.4.

Proof.

(of Theorem 1.4) First note that by the assumption $\Delta\rho\leq 0$ we have in the weak sense that

\Delta\left(\frac{1}{2\rho^{2}}\right)=\text{div}\left(-\frac{\nabla\rho}{\rho^{3}}\right)=-\frac{\Delta\rho}{\rho^{3}}+\frac{3|\nabla\rho|^{2}}{\rho^{4}}\geq\frac{3|\nabla\rho|^{2}}{\rho^{4}}.

Therefore, for any $w\in C^{\infty}_{0}(\mathcal{M},\mathbf{R})$ we have

		$\displaystyle 3\int_{\mathcal{M}}\frac{\|w\|^{2}\|\nabla\rho\|^{2}}{\rho^{4}}dv_{g}\leq\int_{\mathcal{M}}\|w\|^{2}\Delta\left(\frac{1}{2\rho^{2}}\right)dv_{g}$
	$\displaystyle=$	$\displaystyle-2\int_{\mathcal{M}}w\nabla w\cdot\nabla\left(\frac{1}{2\rho^{2}}\right)dv_{g}=2\int_{\mathcal{M}}\left(\|\nabla w\|^{2}+w\Delta w\right)\left(\frac{1}{2\rho^{2}}\right)dv_{g}$
(3.2)		$\displaystyle\leq$	$\displaystyle\left(\int_{\mathcal{M}}\frac{3\|w\|^{2}\|\nabla\rho\|^{2}}{\rho^{4}}dv_{g}\right)^{1/2}\left(\int_{\mathcal{M}}\frac{\|\Delta w\|^{2}}{3\|\nabla\rho\|^{2}}dv_{g}\right)^{1/2}+\int_{\mathcal{M}}\frac{\|\nabla w\|^{2}}{\rho^{2}}dv_{g}\,.$

By applying the first-order Hardy inequality (Theorem 3.2) to $|\nabla w|$ , and using the fact that $|\nabla\rho|\geq\delta$ a.e. on $\mathcal{M}$ , along with Kato’s inequality, Reilly’s formula, and the Poincaré inequality, we obtain:

\int_{\mathcal{M}}\frac{|\nabla w|^{2}}{\rho^{2}}dv_{g}\leq\frac{1}{\delta^{2}}\int_{\mathcal{M}}\frac{|\nabla w|^{2}}{\rho^{2}}|\nabla\rho|^{2}dv_{g}\leq\frac{4}{\delta^{2}}\int_{\mathcal{M}}|\nabla^{2}w|^{2}dv_{g}\leq\frac{C(\mathcal{M})}{\delta^{2}}\int_{\mathcal{M}}|\Delta w|^{2}dv_{g}\,.

Inserting this back to (3) we get

\displaystyle\int_{\mathcal{M}}\frac{|w|^{2}|\nabla\rho|^{2}}{\rho^{4}}dv_{g}\leq C(\mathcal{M})(1+\delta^{-2})^{2}\int_{\mathcal{M}}|\Delta w|^{2}dv_{g}

for some $C(\mathcal{M})>0$ . ∎

4. Proof of the main result

In this section, we prove Theorem 1.2. Let us first fix some notations. Since $\mathcal{M}$ is compact and locally conformally flat with a smooth boundary $\partial\mathcal{M}$ , we can choose a smooth atlas

\{\Phi_{i}:B_{2r_{i}}(p_{i})\to\mathbf{R}^{4}\}_{i=1}^{N}

for $\mathcal{M}$ such that $\{p_{i}\}_{i=1}^{k}\subset\mathcal{M},\{p_{i}\}_{i=k+1}^{N}\subset\partial\mathcal{M}$ and $B_{2r_{i}}(p_{i})$ is conformally flat for each $i$ . We assume that

(4.1)

g=e^{2\phi_{i}}\bar{g}\quad\text{on }B_{2r_{i}}(p_{i})\,,

where $\bar{g}$ is the Euclidean metric. Here $B_{2r_{i}}(p_{i})\subset\mathcal{M}$ denotes the set

B_{2r_{i}}(p_{i}):=\{y\in\mathcal{M}\cup\partial\mathcal{M}:\text{dist}_{g}(y,p_{i})<2r_{i}\},\quad i=1,\cdots,N.

Moreover,

\mathcal{M}\cup\partial\mathcal{M}\text{ is covered by }\left\{B_{r_{i}}(p_{i})\right\}_{i=1}^{N}

and every point in $\mathcal{M}\cup\partial\mathcal{M}$ is covered by $\left\{B_{2r_{i}}(p_{i})\right\}_{i=1}^{N}$ at most $L$ times, see e.g. [41, Lemma 3.3]. For $i=1,\cdots,N$ , let

U_{i}:=\Phi_{i}(B_{r_{i}}(p_{i}))\quad\text{and}\quad 2U_{i}:=\Phi_{i}(B_{2r_{i}}(p_{i}))\,.

When choosing the smooth atlas, we additionally require that there exists $\delta>0$ depending only on $\mathcal{M}$ and $\partial\mathcal{M}$ such that for any $i=k+1,\cdots,N$ and any point $x\in\{y\in\mathcal{M}:\text{dist}(y,\partial\mathcal{M})\leq\delta\}\cap B_{r_{i}}(p_{i})$ we have

(4.2)

\Phi_{i}(B_{\rho(x)}(x))\subset U_{j}\quad\text{for some }j=k+1,\cdots,N\,,

where $\rho(x)=\text{dist}_{g}(x,\partial\mathcal{M})$ ¹¹1To see this, let $\Sigma_{i}=\partial B_{r_{i}}(p_{i})\cap\partial\mathcal{M},i=k+1,\cdots,N$ be the projection of $B_{r_{i}}(p_{i})$ onto $\partial M$ by normal geodesics. $\Gamma_{i}=\partial\Sigma_{i}$ denotes its boundary in $\partial\mathcal{M}$ . We choose the smooth atlas of $\mathcal{M}$ in such a way that every $\Sigma_{i}$ is covered by finitely many $\Sigma_{j_{\alpha}}$ ’s and all the intersecting points $q_{st}=\Gamma_{s}\cap\Gamma_{t}\cap\Sigma_{i}$ stay in the interior of $\Sigma_{l}$ , where $\alpha=1,\dots,\alpha_{i}$ , $j_{\alpha}\in[k+1,N]$ , $l\neq s,t$ and $s,t,l=i,j_{1},\dots,j_{\alpha_{i}}$ . Then there exists a constant $C_{i}>0$ such that for any $y_{0}\in\Sigma_{i}$ , there holds $\text{dist}(y_{0},\Gamma_{l})\geq C_{i}$ for some $l=i,j_{1},\dots,j_{k_{i}}$ . Now, for any $x\in\{y\in\mathcal{M}:\text{dist}_{g}(y,\partial\mathcal{M})\leq\delta\}$ , there exists $x_{0}\in\partial\mathcal{M}$ such that $x_{0}\in\Sigma_{i}$ for some $i\in[k+1,N]$ and $\rho(x)=\text{dist}_{g}(x,\partial\mathcal{M})=\text{dist}_{g}(x,x_{0})$ . Since $B_{\rho(x)}(x)\subset B_{\delta}(x)\subset B_{2\delta}(x_{0})$ , we have $B_{\rho(x)}(x)\subset B_{r_{i}}(p_{i})$ for some $i=k+1,\dots,N$ if we choose $2\delta<\min\limits_{i=k+1,\dots,N}\{C_{i}\}$ .. Moreover,

(4.3)

\left\{y\in\mathcal{M}:\text{dist}(y,\partial\mathcal{M})\geq\delta\right\}\subset\bigcup_{i=1,\cdots,k}B_{r_{i}}(p_{i})\,.

Let $\Omega_{1}:=B_{r_{1}}(p_{1})$ and

(4.4)

\Omega_{i+1}:=\left(\bigcup_{j=1,\cdots,i+1}B_{r_{j}}(p_{j})\right)\setminus\left(\bigcup_{j=1,\cdots,i}\Omega_{j}\right),\quad i=1,\cdots,N-1

be a disjoint partition of $\mathcal{M}$ . Now for any $u\in W^{2,2}(\mathcal{M},\mathbf{S}^{n})$ we define

(4.5)

\bar{u}_{i}(x):=u\left(\Phi_{i}^{-1}(x)\right)\,,\quad x\in 2U_{i},\quad i=1,\cdots,N\,.

Remark 4.1.

Note that by Proposition A.1 we know that if $u$ is a conformal-harmonic map on $\mathcal{M}$ , then $\bar{u}_{i}$ defined in (4.5) is an intrinsic biharmonic map on $2U_{i},i=1,\cdots,N$ .

In what follows, we will denote $|\cdot|$ the norm with respect to the Euclidean metric in $\mathbf{R}^{K}$ . As before, $\bar{\nabla}$ and $\bar{\Delta}$ will denote the connection and Laplacian with respect to the Euclidean metric $\bar{g}$ , but note that on each $\Omega_{i},i=1,\cdots,N$ the information of $\Phi_{i}$ is embedded in these two operators and we do not differentiate the notations.

Lemma 4.2.

There exists $\epsilon_{0}>0$ depending only on $\mathcal{M}$ such that if $u,v$ are as in Theorem 1.2, then we have

(4.6)

\int_{\mathcal{M}}|\bar{\Delta}(v-u)|^{2}dv_{\bar{g}}\leq 4\int_{\mathcal{M}}|\bar{\tau}(v)-\bar{\tau}(u)|^{2}dv_{\bar{g}}\,,

where $\bar{\tau}(u)=(\bar{\Delta}u)^{T}$ is the tension field of $u$ with respect to the flat connection.

Proof.

Let $\bar{u}_{i},\bar{v}_{i}$ be defined as in (4.5), then use the conformal change of Laplacian (c.f. (A.6)), for $i=1,\cdots,N$ we get

	$\displaystyle\int_{2U_{i}}\|\bar{\Delta}\bar{u}_{i}\|^{2}dx$	$\displaystyle=\int_{B_{2r_{i}}(p_{i})}\|\bar{\Delta}u\|^{2}dv_{\bar{g}}$
(4.7)			$\displaystyle\leq C_{i}\int_{B_{2r_{i}}(p_{i})}(\|\Delta u\|^{2}+\|\nabla u\|^{2})dv_{g}\leq C_{i}\sqrt{\epsilon_{0}}$

and

(4.8)

\int_{2U_{i}}|\bar{\nabla}\bar{v}_{i}|^{4}dx=\int_{B_{2r_{i}}(p_{i})}|\bar{\nabla}v|^{4}dv_{\bar{g}}=\int_{B_{2r_{i}}(p_{i})}|\nabla v|^{4}dv_{g}\leq\epsilon_{0}\,,

where $C_{i}=C_{i}(\Phi_{i},\phi_{i})>0$ are positive constants. Moreover, using (4), (4.8), (1.5) and

\int_{\mathcal{M}}|\Delta u|^{2}dv_{g}=\int_{\mathcal{M}}|\tau(u)|^{2}dv_{g}+\int_{\mathcal{M}}|\nabla u|^{4}dv_{g}\,,

we have

(4.9)

\int_{\mathcal{M}}|\bar{\Delta}u|^{2}dv_{\bar{g}}\leq C\int_{\mathcal{M}}\left(|\Delta u|^{2}+|\nabla u|^{2}\right)dv_{g}\leq C\sqrt{\epsilon_{0}}

and

(4.10)

\int_{\mathcal{M}}|\bar{\nabla}v|^{4}dv_{\bar{g}}=\int_{\mathcal{M}}|\nabla v|^{4}dv_{g}\leq\epsilon_{0}\,,

where $C>0$ may depend on all $\phi_{i}$ and $\Phi_{i},i=1,\cdots,N$ . Now using the decomposition $\bar{\Delta}u=\bar{\tau}(u)-u|\bar{\nabla}u|^{2}$ , we have

		$\displaystyle\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}=\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)-v\|\bar{\nabla}v\|^{2}+u\|\bar{\nabla}u\|^{2}\|^{2}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+2\int_{\mathcal{M}}\left\|v\|\bar{\nabla}v\|^{2}-u\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+2\int_{\mathcal{M}}\left\|v\left(\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2}\right)+(v-u)\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\left\|\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+4\left(\int_{\mathcal{M}}\|\bar{\nabla}(v+u)\|^{4}dv_{\bar{g}}\right)^{\frac{1}{2}}\left(\int_{\mathcal{M}}\|\bar{\nabla}(v-u)\|^{4}dv_{\bar{g}}\right)^{\frac{1}{2}}dv_{\bar{g}}$
		$\displaystyle+4\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+C\sqrt{\epsilon_{0}}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}\,,$

where we have used (1.5), (4.9), (4.10),

(4.11)

\int_{\mathcal{M}}|\bar{\nabla}u|^{4}dv_{\bar{g}}\leq\int_{\mathcal{M}}|\bar{\Delta}u|^{2}dv_{\bar{g}}\leq C\sqrt{\epsilon_{0}}

and

(4.12)

\left(\int_{\mathcal{M}}|\bar{\nabla}(v-u)|^{4}dv_{\bar{g}}\right)^{\frac{1}{4}}\leq C\left(\int_{\mathcal{M}}|\bar{\Delta}(v-u)|^{2}dv_{\bar{g}}\right)^{\frac{1}{2}}\,,

where $C>0$ is a universal constant. For the last term above, we claim that

(4.13)

\int_{\mathcal{M}}|v-u|^{2}|\bar{\nabla}u|^{4}dv_{\bar{g}}\leq C_{0}{\epsilon_{0}}\int_{\mathcal{M}}|v-u|^{2}\rho^{-4}dv_{g}\,,

where $\rho(x)=\text{dist}_{g}(x,\partial\mathcal{M})$ and $C_{0}=C_{0}(\mathcal{M})>0$ . If the claim (4.13) is true, then the second Hardy inequality (Theorem 1.5) imply that (since $v-u\in W^{2,2}_{0}(\mathcal{M},\mathbf{S}^{n})$ and $\frac{\partial(v-u)}{\partial\nu}=0$ on $\partial\mathcal{M}$ )

		$\displaystyle\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}\leq C{\epsilon_{0}}\int_{\mathcal{M}}\|\Delta(v-u)\|^{2}dv_{g}$
	$\displaystyle=$	$\displaystyle C{\epsilon_{0}}\sum_{i=1,\cdots,N}\int_{\Omega_{i}}\|\Delta(v-u)\|^{2}dv_{g}$
	$\displaystyle\leq$	$\displaystyle C{\epsilon_{0}}\sum_{i=1,\cdots,N}\int_{\Omega_{i}}\left(\|\bar{\Delta}(v-u)\|^{2}+\|\bar{\nabla}(v-u)\|^{2}\right)dv_{\bar{g}}$
(4.14)		$\displaystyle=$	$\displaystyle C{\epsilon_{0}}\int_{\mathcal{M}}\left(\|\bar{\Delta}(v-u)\|^{2}+\|\bar{\nabla}(v-u)\|^{2}\right)dv_{\bar{g}}\leq C{\epsilon_{0}}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}\,.$

where we used $\int_{\Omega_{i}}|\Delta(v-u)|^{2}dv_{g}\leq C_{i}\int_{\Omega_{i}}\left(|\bar{\Delta}(v-u)|^{2}+|\bar{\nabla}(v-u)|^{2}\right)dv_{\bar{g}}$ for each $i$ , which is similar to (4). Therefore, we obtain (4.6) by choosing $\epsilon_{0}$ sufficiently small. To complete the proof of this lemma, it remains to show that the claim (4.13) holds.

Using (4), Remark 4.1 and Theorem 2.2, and choosing $\epsilon_{0}$ small enough we have

(4.15)

|\bar{\nabla}^{l}\bar{u}_{i}|(x)\leq C(\text{dist}_{\bar{g}}(x,\partial(2U_{i})))^{-l}{\epsilon_{0}}^{1/4}\,,

for any $x\in 2U_{i},i=1,\cdots,k$ and $l=1,2,3$ , where $C>0$ depends on the $C_{i}$ above. In particular, define

r_{0}:=\min_{i=1,\cdots,k}\text{dist}_{\bar{g}}(U_{i},\partial(2U_{i}))\,,

then we have

(4.16)

|\bar{\nabla}^{l}\bar{u}_{i}|(x)\leq Cr_{0}^{-l}{\epsilon_{0}}^{1/4}\quad\text{for any }x\in U_{i},i=1,\cdots,k\text{ and }l=1,2,3,

(4.17)

|\bar{\nabla}^{l}u|(x)\leq Cr_{0}^{-l}{\epsilon_{0}}^{1/4}\quad\text{for any }x\in B_{r_{i}}(p_{i}),i=1,\cdots,k\text{ and }l=1,2,3,

and therefore

(4.18)

|\bar{\nabla}^{l}u|(x)\leq Cr_{0}^{-l}{\epsilon_{0}}^{1/4}(\text{diam}_{g}(\mathcal{M}))^{l}\rho^{-l}(x)\leq C_{1}{\epsilon_{0}}^{1/4}\rho^{-l}(x)

for any $x\in B_{r_{i}}(p_{i}),i=1,\cdots,k$ and $l=1,2,3$ , where the $C>0$ in (4.17) and (4.18) may depend on $\Phi_{i},i=1,\cdots,k,$ and

C_{1}:=\max_{l=1,2,3}\left\{Cr_{0}^{-l}(\text{diam}_{g}(\mathcal{M}))^{l}\right\}\,.

Now by the choice of the smooth atlas, in particular, (4.2), with a similar argument as above, we have

(4.19)

|\bar{\nabla}^{l}u|(x)\leq C_{2}{\epsilon_{0}}^{1/4}\rho^{-l}(x)

for any $x\in B_{r_{i}}(p_{i})\cap\{y\in\mathcal{M}:\text{dist}(y,\partial\mathcal{M})\leq\delta\},i=k+1,\cdots,N$ and $l=1,2,3$ . Here $C_{2}>0$ may depend on all $\Phi_{i}$ and $\phi_{i},i=k,\cdots,N.$ Then with a argument similar to (4) we get (4.13). This completes the proof of the lemma. ∎

Lemma 4.3.

There exists $\epsilon_{0}>0$ depending only on $\mathcal{M}$ such that if $u,v$ are as in Theorem 1.2, then we have

		$\displaystyle\int_{\mathcal{M}}\|\bar{\Delta}v\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\Delta}u\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}$
(4.20)		$\displaystyle\geq$	$\displaystyle-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}\bar{\nabla}u\cdot\bar{\nabla}(v-u)dv_{\bar{g}}\,.$

Proof.

By Remark 4.1, we know that $u$ is an intrinsic biharmonic map on each $\Omega_{i},i=1,\cdots,N,$ with respect to the flat connection $\bar{\nabla}$ , namely, we have

	$\displaystyle\bar{\Delta}^{2}u=$	$\displaystyle P(\bar{\Delta}^{2}u)+P\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,{D}_{u}({D}_{u}P^{\bot})\bar{\nabla}u\bar{\nabla}u\rangle$
		$\displaystyle-2{\rm div}_{\bar{g}}\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}P\rangle+2\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}\bar{\nabla}P\rangle\,.$

Since $u=v$ , $\partial_{\nu}u=\partial_{\nu}v$ on $\partial\mathcal{M}$ , we have

		$\displaystyle\int_{\mathcal{M}}\|\bar{\Delta}v\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\Delta}u\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\left\langle\bar{\Delta}^{2}u,v-u\right\rangle dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\left\langle P^{\bot}(\bar{\Delta}^{2}u),v-u\right\rangle dv_{\bar{g}}+2\int_{\mathcal{M}}\left\langle P(\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,{D}_{u}({D}_{u}P^{\bot})\bar{\nabla}u\bar{\nabla}u\rangle),v-u\right\rangle dv_{\bar{g}}$
		$\displaystyle-4\int_{\mathcal{M}}\left\langle{\rm div}_{\bar{g}}\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}P\rangle,v-u\right\rangle dv_{\bar{g}}+4\int_{\mathcal{M}}\left\langle\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}\bar{\nabla}P\rangle,v-u\right\rangle dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\left\langle P^{\bot}(\bar{\Delta}^{2}u),v-u\right\rangle dv_{\bar{g}}+2\int_{\mathcal{M}}\left\langle\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,{D}_{u}({D}_{u}P^{\bot})\bar{\nabla}u\bar{\nabla}u\rangle,P(v-u)\right\rangle dv_{\bar{g}}$
		$\displaystyle+4\int_{\mathcal{M}}\left\langle\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}P\rangle,\bar{\nabla}(v-u)\right\rangle dv_{\bar{g}}+4\int_{\mathcal{M}}\left\langle\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}\bar{\nabla}P\rangle,v-u\right\rangle dv_{\bar{g}}$
	$\displaystyle=:$	$\displaystyle\textbf{I}+\textbf{II}+\textbf{III}+\textbf{IV}.$

For term I, we can use 1.5, a similar argument as in the proof of (4.13) and the second order Hardy inequality (Theorem 1.5) to get

	$\displaystyle 2\int_{\mathcal{M}}\left\langle P^{\bot}(\bar{\Delta}^{2}u),v-u\right\rangle dv_{\bar{g}}$	$\displaystyle\geq-C\int_{\mathcal{M}}\left\|(v-u)^{\bot}\right\|\cdot\left\|P^{\bot}(\Delta^{2}u)\right\|dv_{\bar{g}}$
		$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|v-u\|^{2}\rho(x)^{-4}dv_{g}$
		$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\Delta(v-u)\|^{2}dv_{g}$
(4.21)			$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}\,.$

For term II, we note that the integrand reads as

\sum_{\alpha,\beta,\gamma,i,j,k,m,l}P_{lk}\bar{\nabla}_{\alpha}(P^{\bot})_{ij}\bar{\nabla}_{\alpha}u^{j}D_{u^{k}}D_{u^{\beta}}(P^{\bot})_{im}\bar{\nabla}_{\gamma}u^{\beta}\bar{\nabla}_{\gamma}u^{m}(v-u)^{l}\,.

Now since the target manifold is $\mathbf{S}^{n}$ , we know that $u$ is the unit normal vector at the point $u\in\mathbf{S}^{n}$ and $P^{\bot}({\mathbf{v}})=\langle{\mathbf{v}},u\rangle u$ for any vector ${\mathbf{v}}\in T_{u}(\mathbf{R}^{n+1})$ so that

P^{\bot}_{ij}=u^{i}u^{j}\quad\text{and}\quad P^{\bot}(\bar{\Delta}u)=-\bar{\nabla}P^{\bot}\bar{\nabla}u=-u|\bar{\nabla}u|^{2}\,.

Therefore, in this case the integrand in term II becomes

(4.22)

\sum_{\beta,\gamma,i,k,m,l}P_{lk}u^{i}|\nabla u|^{2}(\delta_{i\beta}\delta_{mk}+\delta_{ik}\delta_{m\beta})\bar{\nabla}_{\gamma}u^{\beta}\bar{\nabla}_{\gamma}u^{m}(v-u)^{l}\,=\,0\,.

For term III, we have (using again $P^{\bot}_{ij}=u^{i}u^{j}$ )

(4.23)

\left\langle\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}P\rangle,\bar{\nabla}(v-u)\right\rangle=|\bar{\nabla}u|^{2}\bar{\nabla}u\cdot\bar{\nabla}(v-u).

For term IV, we use again $P^{\bot}_{ij}=u^{i}u^{j},P+P^{\bot}=\text{Id}$ and also $u\cdot\bar{\nabla}u=0$ on $\mathbf{S}^{n}$ to get

		$\displaystyle\int_{\mathcal{M}}\left\langle\langle\bar{\nabla}P^{\bot}\bar{\nabla}u,\bar{\nabla}P^{\bot}\bar{\nabla}P\rangle,v-u\right\rangle dv_{\bar{g}}=\int_{\mathcal{M}}\bar{\nabla}P^{\bot}_{ij}\bar{\nabla}u^{j}\bar{\nabla}P^{\bot}_{ik}\bar{\nabla}P_{ks}(v-u)^{s}dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle-\int_{\mathcal{M}}\|\bar{\nabla}u\|^{4}u\cdot(v-u)dv_{\bar{g}}\geq-2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{4}\|(v-u)^{\bot}\|dv_{\bar{g}}$
	$\displaystyle\geq$	$\displaystyle-C\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}\,.$

Then by (4.13) and (4) we have

(4.24)

\textbf{IV}\geq-C\epsilon_{0}^{1/4}\int_{\mathcal{M}}|\bar{\Delta}(v-u)|^{2}dv_{\bar{g}}\,.

Now (4.3) follows directly by combining the estimates above. ∎

Lemma 4.4.

There exists $\epsilon_{0}>0$ depending only on $\mathcal{M}$ such that if $u,v$ are as in Theorem 1.2, then we have

		$\displaystyle\int_{\mathcal{M}}\|\bar{\tau}(v)\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\tau}(u)\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}$
(4.25)		$\displaystyle\geq$	$\displaystyle-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}\,.$

Proof.

It follows from Lemma 4.3 and (4.3) that

	$\displaystyle\psi=$	$\displaystyle\int_{\mathcal{M}}\|\bar{\tau}(v)\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\tau}(u)\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle\int_{\mathcal{M}}\|\bar{\Delta}v\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\Delta}u\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}$
		$\displaystyle-\int_{\mathcal{M}}\|\bar{\nabla}v\|^{4}-\|\bar{\nabla}u\|^{4}dv_{\bar{g}}+\int_{\mathcal{M}}\left\|v\|\bar{\nabla}v\|^{2}+u\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
		$\displaystyle+2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}u\bar{\Delta}v+\|\bar{\nabla}v\|^{2}v\bar{\Delta}u\,dv_{\bar{g}}$
	$\displaystyle\geq$	$\displaystyle-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}\bar{\nabla}u\bar{\nabla}(v-u)dv_{\bar{g}}$
		$\displaystyle-\int_{\mathcal{M}}\|\bar{\nabla}v\|^{4}-\|\bar{\nabla}u\|^{4}dv_{\bar{g}}+\int_{\mathcal{M}}\left\|v\|\bar{\nabla}v\|^{2}+u\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
		$\displaystyle+2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}u\bar{\Delta}v+\|\bar{\nabla}v\|^{2}v\bar{\Delta}u\,dv_{\bar{g}}.$

Therefore, using $u|_{\partial\mathcal{M}}=v|_{\partial\mathcal{M}}$ and $\partial_{\nu}u|_{\partial\mathcal{M}}=\partial_{\nu}v|_{\partial\mathcal{M}}$ , we have

	$\displaystyle\psi\geq$	$\displaystyle-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}-2\int_{\mathcal{M}}\bar{\nabla}\|\bar{\nabla}u\|^{2}u\,\bar{\nabla}vdv_{\bar{g}}-2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}u\bar{\Delta}v\,dv_{\bar{g}}$
		$\displaystyle-2\int_{\mathcal{M}}\bar{\nabla}\|\bar{\nabla}u\|^{2}v\bar{\nabla}u\,dv_{\bar{g}}-2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}v\bar{\Delta}u\,dv_{\bar{g}}-4\int_{\mathcal{M}}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}$
		$\displaystyle-\int_{\mathcal{M}}(\|\bar{\nabla}v\|^{4}-\|\bar{\nabla}u\|^{4})dv_{\bar{g}}+\int_{\mathcal{M}}\left\|v\|\bar{\nabla}v\|^{2}+u\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
		$\displaystyle+2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}u\bar{\Delta}v+\|\bar{\nabla}v\|^{2}v\bar{\Delta}u\,dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+\int_{\mathcal{M}}\bar{\nabla}\|\bar{\nabla}u\|^{2}\bar{\nabla}\|v-u\|^{2}dv_{\bar{g}}$
		$\displaystyle+2\int_{\mathcal{M}}v\bar{\Delta}u(\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2})dv_{\bar{g}}+2\int_{\mathcal{M}}\|\bar{\nabla}u\|^{2}uv(\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2})dv_{\bar{g}}$
		$\displaystyle-\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}\,,$

where we have used $|u|^{2}=|v|^{2}=1$ so that $1-uv=\frac{1}{2}|v-u|^{2}$ and $u\bar{\nabla}v+v\bar{\nabla}u=-\frac{1}{2}\bar{\nabla}|v-u|^{2}$ . Thus, we have

	$\displaystyle\psi$	$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+2\int_{\mathcal{M}}(\bar{\Delta}u+u\|\bar{\nabla}u\|^{2})v(\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2})dv_{\bar{g}}$
		$\displaystyle=-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+2\int_{\mathcal{M}}\bar{\tau}(u)\cdot(v-u)\langle\bar{\nabla}(v+u),\bar{\nabla}(v-u)\rangle\;dv_{\bar{g}}$
		$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}$
		$\displaystyle\quad-2\left(\int_{\mathcal{M}}\|\bar{\tau}(u)\|^{2}\|v-u\|^{2}dv_{\bar{g}}\right)^{\frac{1}{2}}\left(\int_{\mathcal{M}}\|\bar{\nabla}(v+u)\|^{4}dv_{\bar{g}}\right)^{\frac{1}{4}}\left(\int_{\mathcal{M}}\|\bar{\nabla}(v-u)\|^{4}dv_{\bar{g}}\right)^{\frac{1}{4}}$
		$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}\;dv_{\bar{g}}\,,$

where we used (4.12), (4.18), (4.19), Hardy inequality (Theorem 1.5), (4.10) and (4.11). ∎

Proof of Theorem 1.2.

From (4), we have

(4.26)

\int_{\mathcal{M}}|\Delta v-\Delta u|^{2}dv_{g}\leq C\int_{\mathcal{M}}|\bar{\Delta}v-\bar{\Delta}u|^{2}dv_{\bar{g}}\,.

Combining (4.6) and (4.4) we get

		$\displaystyle\int_{\mathcal{M}}\|\bar{\tau}(u)\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\tau}(v)\|^{2}dv_{\bar{g}}+\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}$
(4.27)		$\displaystyle\leq$	$\displaystyle\,C\epsilon_{0}^{1/4}\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}\,.$

Now choosing $\epsilon_{0}$ sufficiently small we get (using (4.26) and (4))

		$\displaystyle\int_{\mathcal{M}}\|\Delta v-\Delta u\|^{2}dv_{g}\leq C\int_{\mathcal{M}}\|\bar{\Delta}v-\bar{\Delta}u\|^{2}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 4C\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}\leq 8C\left(\int_{\mathcal{M}}\|\bar{\tau}(v)\|^{2}dv_{\bar{g}}-\int_{\mathcal{M}}\|\bar{\tau}(u)\|^{2}dv_{\bar{g}}\right)$
	$\displaystyle=$	$\displaystyle 8C\sum_{i=1,\cdots,N}\left(\int_{\Omega_{i}}\|\bar{\tau}(v)\|^{2}dv_{\bar{g}}-\int_{\Omega_{i}}\|\bar{\tau}(u)\|^{2}dv_{\bar{g}}\right)=8C\sum_{i=1,\cdots,N}\left(\mathcal{E}(v,\Omega_{i})-\mathcal{E}(u,\Omega_{i})\right)$
	$\displaystyle=$	$\displaystyle 8C\left(\mathcal{E}(v,\mathcal{M})-\mathcal{E}(u,\mathcal{M})\right)\,.$

This completes the proof of Theorem 1.2. ∎

5. Conflict of interest and Data availability

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. Data sharing not applicable to this article as no datasets were generated or analyzed in this study.

Appendix A Conformal invariance of $\mathcal{E}(u,\mathcal{M})$

In this appendix we give a quick check of the conformal invariance of $\mathcal{E}(u,\mathcal{M})$ in four dimensions which does not seem to be in the literature. Let $\mathcal{M}$ be a smooth Riemannian 4-manifold with or without boundary and $\mathcal{N}$ be another smooth Riemannian manifold. Given $u\in C^{2}(\mathcal{M},\mathcal{N})$ , we denote by $u^{*}T\mathcal{N}$ the pull-back on $(\mathcal{M},g)$ of the tangent bundle of $(\mathcal{N},h)\hookrightarrow\mathbf{R}^{K}$ . Then the tension field $\tau(u)\in u^{*}T\mathcal{N}$ is defined by

(A.1)

\tau(u)=\sum_{\alpha}{\tilde{\nabla}}_{e_{\alpha}}du(e_{\alpha}),

where $\{e_{\alpha}\}$ is an orthonormal frame of $T\mathcal{M}$ and $\tilde{\nabla}$ is the metric connection on $T^{*}\mathcal{M}\otimes u^{*}T\mathcal{N}$ . Fix local coordinates $\{x^{\alpha}\}$ and $\{y^{i}\}$ of $\mathcal{M}$ and $\mathcal{N}$ respectively, we have

(A.2)

\tau^{i}(u)=g^{\alpha\beta}\frac{\partial^{2}u^{i}}{\partial x^{\alpha}\partial x^{\beta}}-g^{\alpha\beta}\Gamma^{\gamma}_{\alpha\beta}\frac{\partial u^{i}}{\partial x^{\gamma}}+g^{\alpha\beta}\bar{\Gamma}^{i}_{jk}\frac{\partial u^{j}}{\partial x^{\alpha}}\frac{\partial u^{k}}{\partial x^{\beta}},

where $\Gamma$ and $\bar{\Gamma}$ are the Christoffel symbols on $\mathcal{M}$ and $\mathcal{N}$ respectively. Define

A^{i}:=g^{\alpha\beta}\frac{\partial^{2}u^{i}}{\partial x^{\alpha}\partial x^{\beta}}+g^{\alpha\beta}\bar{\Gamma}^{i}_{jk}\frac{\partial u^{j}}{\partial x^{\alpha}}\frac{\partial u^{k}}{\partial x^{\beta}}.

Then

(A.3)

\begin{split}\tau^{i}(u)&=A^{i}-g^{\alpha\beta}\Gamma^{\gamma}_{\alpha\beta}\frac{\partial u^{i}}{\partial x^{\gamma}}=A^{i}-g^{\alpha\beta}\langle\nabla_{X_{\alpha}}X_{\beta},du^{i}\rangle,\end{split}

where $\nabla=\nabla_{g}$ is the connection on $\mathcal{M}$ and $X_{\alpha}:=\frac{\partial}{\partial x^{\alpha}}$ . Now under the conformal change $\bar{g}=e^{2\phi}g$ (with $\phi\in C^{\infty}_{0}(\mathcal{M},\mathbf{R})$ if $\partial\mathcal{M}\neq\emptyset$ ), we have

(A.4)

\bar{A}^{i}=e^{-2\phi}A^{i}

and

(A.5)

\bar{g}^{\alpha\beta}\langle(\bar{\nabla}_{\bar{g}})_{X_{\alpha}}X_{\beta},du^{i}\rangle=e^{-2\phi}g^{\alpha\beta}\langle\nabla_{X_{\alpha}}X_{\beta},du^{i}\rangle-2e^{-2\phi}\langle\nabla\phi,du^{i}\rangle\,,

where we have used

(A.6)

(\bar{\nabla}_{\bar{g}})_{X}Y=\nabla_{X}Y+X(\phi)Y+Y(\phi)X-g(X,Y)\nabla\phi.

Therefore, combing (A.4) and (A.5) together, we get

(A.7)

\bar{\tau}^{i}(u)=e^{-2\phi}\tau^{i}(u)+2e^{-2\phi}\langle\nabla\phi,du^{i}\rangle

and

(A.8)

|\bar{\tau}(u)|^{2}-e^{-4\phi}|\tau(u)|^{2}=4e^{-4\phi}h_{ij}(u)\tau^{j}(u)\langle\nabla\phi,du^{i}\rangle+4e^{-4\phi}|\langle\nabla\phi,du\rangle|^{2}.

Recall the following formulas for the conformal change of Ricci tensor and scalar curvature:

(A.9)

\overline{\rm Ric}={\rm Ric}-2({\rm Hess}(\phi)-\nabla\phi\otimes\nabla\phi)-(\Delta\phi+2|\nabla\phi|^{2})g

and

(A.10)

\overline{\rm{Sc}}=e^{-2\phi}({\rm{Sc}}-6\Delta\phi-6|\nabla\phi|^{2}),

where ${\rm Hess}(\phi)$ denotes the Hessian of $\phi$ . Consequently, we obtain

(A.11)		$\displaystyle\frac{2}{3}\overline{\rm{Sc}}\|du\|_{\bar{g}}^{2}-\frac{2}{3}e^{-4\phi}{\rm{Sc}}\|du\|^{2}_{g}$	$\displaystyle=\frac{2}{3}e^{-4\phi}(-6\Delta\phi-6\|\nabla\phi\|^{2})\|du\|^{2}_{g}$
		$\displaystyle=-4e^{-4\phi}(\Delta\phi+\|\nabla\phi\|^{2})\|du\|^{2}_{g}$

and

		$\displaystyle 2\overline{\rm Ric}(du,du)-2e^{-4\phi}{\rm Ric}(du,du)$
	$\displaystyle=$	$\displaystyle 2\bar{R}_{\alpha\beta}\bar{g}^{\alpha\gamma}\bar{g}^{\beta\sigma}\left\langle\frac{\partial u}{\partial x_{\gamma}},\frac{\partial u}{\partial x_{\sigma}}\right\rangle-2e^{-4\phi}{\rm Ric}(du,du)$
	$\displaystyle=$	$\displaystyle 2e^{-4\phi}g^{\alpha\gamma}g^{\beta\sigma}\bigg{(}(-\Delta\phi-2\|\nabla\phi\|^{2})g_{\alpha\beta}-2(\nabla_{\beta}\phi_{\alpha}-\phi_{\alpha}\phi_{\beta})\bigg{)}\langle u_{\gamma},u_{\sigma}\rangle$
(A.12)		$\displaystyle=$	$\displaystyle-4e^{-4\phi}\|\nabla\phi\|^{2}\|du\|_{g}^{2}+4e^{-4\phi}\|\langle\nabla\phi,du\rangle\|^{2}-2e^{-4\phi}\Delta\phi\|du\|_{g}^{2}$
		$\displaystyle-4e^{-4\phi}\sum_{\alpha,\beta}\nabla^{2}_{\alpha\beta}\phi\left\langle\nabla_{\alpha}u,\nabla_{\beta}u\right\rangle\,,$

where Einstein summation convention was used and we used different subscripts to distinguish different metrics on the bundle $T^{*}\mathcal{M}\otimes u^{*}T\mathcal{N}$ . Combining these we have

		$\displaystyle\int_{\mathcal{M}}\left[\eqref{tau}+\eqref{s}-\eqref{ric}\right]dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle\sum_{\alpha,\beta}\int_{\mathcal{M}}2e^{-4\phi}\left(2h_{ij}(u)\tau^{j}(u)\langle\nabla\phi,du^{i}\rangle-\Delta\phi\|du\|_{g}^{2}+2\nabla^{2}_{\alpha\beta}\phi\langle\nabla_{\alpha}u,\nabla_{\beta}u\rangle\right)dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle{\sum_{\alpha,\beta}\int_{\mathcal{M}}4\nabla_{\alpha}\phi\langle\Delta u,\nabla_{\alpha}{u}\rangle-2\Delta\phi\|du\|_{g}^{2}+4\nabla^{2}_{\alpha\beta}\phi\langle\nabla_{\alpha}u,\nabla_{\beta}u\rangle dv_{g}}$
	$\displaystyle=$	$\displaystyle\sum_{\alpha,\beta}\int_{\mathcal{M}}-2\Delta\phi\|du\|^{2}-4\nabla_{\beta}\phi\langle\nabla_{\alpha}u,\nabla_{\alpha\beta}^{2}u\rangle dv_{g}$
	$\displaystyle=$	$\displaystyle\sum_{\alpha,\beta}\int_{\mathcal{M}}2\nabla_{\beta}{\phi}\nabla_{\beta}\langle\nabla_{\alpha}u,\nabla_{\alpha}u\rangle-4\nabla_{\beta}\phi\langle\nabla_{\alpha}u,\nabla^{2}_{\alpha\beta}u\rangle dv_{g}=0.$

Hence, the energy functional $\mathcal{E}(u,\mathcal{M})$ is conformally invariant in four dimensions. Now using the notations in (4.1), on each $\Omega_{i},i=1,\cdots,N$ we have

g=e^{2\phi_{i}}\bar{g}\,,

where $\bar{g}$ is the standard Euclidean metric. By the above conformal change of $\mathcal{E}(\cdot,\mathcal{M})$ , for any $w\in C^{2}(\mathcal{M},\mathcal{N})$ we get

		$\displaystyle\mathcal{E}(w,(\mathcal{M},g))$
	$\displaystyle=$	$\displaystyle\sum_{i}\mathcal{E}(w,(\Omega_{i},\bar{g}))+\sum_{i,\alpha,\beta}\int_{\Omega_{i}}4\bar{\nabla}_{\alpha}\phi_{i}\langle\bar{\Delta}w,\bar{\nabla}_{\alpha}w\rangle-2\bar{\Delta}\phi_{i}\|\bar{\nabla}w\|^{2}+4\bar{\nabla}^{2}_{\alpha\beta}\phi_{i}\langle\bar{\nabla}_{\alpha}w,\bar{\nabla}_{\beta}w\rangle dv_{\bar{g}}.$

Thus, for any variation $w(t)$ of $w$ such that $\left.\frac{d}{dt}\right|_{t=0}w(t)=P\phi\in T_{w}\mathcal{N}$ where $\phi\in C^{\infty}_{0}(\mathcal{M},\mathbf{R}^{K})$ , the first variation formula for $\mathcal{E}(w,(\mathcal{M},g))$ is (cf. Appendix B)

(A.13)			$\displaystyle\left.\frac{d}{dt}\right\|_{t=0}{\mathcal{E}}(w(t),(\mathcal{M},g))$
	$\displaystyle=$	$\displaystyle 2\sum_{i}\int_{\Omega_{i}}e^{-4\phi_{i}}\big{\langle}P(\bar{\Delta}^{2}u)-P(A(\bar{\nabla}u,\bar{\nabla}u){D}_{u}A(\bar{\nabla}u,\bar{\nabla}u))$
		$\displaystyle+2{\rm div}(A(\bar{\nabla}u,\bar{\nabla}u)A(\bar{\nabla}u,P))-2A(\bar{\nabla}u,\bar{\nabla}u)A(\bar{\nabla}u,\bar{\nabla}P),\phi\big{\rangle}_{g}dv_{g}$
(A.14)			$\displaystyle+\sum_{i,\alpha,\beta}\int_{\Omega_{i}}\big{[}4\bar{\nabla}_{\alpha}\phi_{i}\langle\bar{\Delta}(P\phi),\bar{\nabla}_{\alpha}w\rangle+4\bar{\nabla}_{\alpha}\phi_{i}\langle\bar{\Delta}w,\bar{\nabla}_{\alpha}(P\phi)\rangle$
(A.15)			$\displaystyle-4\bar{\Delta}\phi_{i}\langle\bar{\nabla}w,\nabla(P\phi)\rangle+8\bar{\nabla}^{2}_{\alpha\beta}\phi_{i}\langle\bar{\nabla}_{\alpha}(P\phi),\bar{\nabla}_{\beta}w\rangle\big{]}dv_{\bar{g}}\,.$

In particular, for any fixed $j=1,\cdots,N$ and any $\phi\in C^{\infty}_{0}(\Omega_{j},\mathbf{R}^{K})$ we have

	$\displaystyle\left.\frac{d}{dt}\right\|_{t=0}{\mathcal{E}}(w(t),(\mathcal{M},g))$
(A.16)	$\displaystyle=$	$\displaystyle 2\int_{\Omega_{j}}e^{-4\phi_{j}}\big{\langle}P(\bar{\Delta}^{2}u)-P(A(\bar{\nabla}u,\bar{\nabla}u){D}_{u}A(\bar{\nabla}u,\bar{\nabla}u))$
	$\displaystyle+2{\rm div}(A(\bar{\nabla}u,\bar{\nabla}u)A(\bar{\nabla}u,P))-2A(\bar{\nabla}u,\bar{\nabla}u)A(\bar{\nabla}u,\bar{\nabla}P),\phi\big{\rangle}_{g}dv_{g}\,,$

noting that since $\phi\in C^{\infty}_{0}(\Omega_{j},\mathbf{R}^{K})$ , the terms in (A.14) and (A.15) cancel out to zero using integration by parts over $\Omega_{j}$ . Therefore we have

Proposition A.1.

If $u$ is a conformal-harmonic map on a locally conformally flat $4$ -manifold $(\mathcal{M},g)$ , then $u$ is locally an intrinsic biharmonic map on $\mathcal{M}$ (with respect to the flat connection) up to a conformal change of $g$ .

Appendix B Conformal-harmonic map equation

For the sake of completeness, in this appendix we include a derivation of the conformal-harmonic map equation (1), i.e., the Euler-Lagrangian equation for $\mathcal{E}(u,\mathcal{M})$ . Consider a variation $u(t)$ of $u$ such that $\left.\frac{d}{dt}\right|_{t=0}u(t)=P\phi\in T_{u}\mathcal{N}$ where $\phi\in C^{\infty}_{0}(\mathcal{M},\mathbf{R}^{K})$ . Then

		$\displaystyle\left.\frac{d}{dt}\right\|_{t=0}\mathcal{E}(u(t),\mathcal{M})$
	$\displaystyle=$	$\displaystyle\int_{\mathcal{M}}\frac{d}{dt}\bigg{\|}_{t=0}\|\tau(u(t))\|^{2}+\frac{2}{3}{\rm Sc}^{\mathcal{M}}\frac{d}{dt}\bigg{\|}_{t=0}\|du(t)\|^{2}-2\frac{d}{dt}\bigg{\|}_{t=0}{\rm Ric}^{\mathcal{M}}(du(t),du(t))$
(B.1)		$\displaystyle:=$	$\displaystyle\,{\textbf{I}}+{\textbf{II}}+{\textbf{III}}.$

We compute the three terms in (B) as follow. Note that the first term I is the first variation of the intrinsic bienergy $\int_{\mathcal{M}}|\tau(u)|^{2}$ and therefore yields exactly the intrinsic biharmonic map equation.

	$\displaystyle{\textbf{I}}=$	$\displaystyle\frac{d}{dt}\bigg{\|}_{t=0}\int_{\mathcal{M}}\left\|\Delta u(t)\|^{2}-\|A(u(t))(\nabla u(t),\nabla u(t))\right\|^{2}$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\langle\Delta u,\Delta(P\phi)\rangle-A(\nabla u,\nabla u)(D_{u}A(\nabla u,\nabla u)P\phi+2A(\nabla u,\nabla(P\phi)))$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\langle\Delta^{2}u,P\phi\rangle-\langle A(\nabla u,\nabla u)D_{u}A(\nabla u,\nabla u),P\phi\rangle$
		$\displaystyle+2\langle\nabla(A(\nabla u,\nabla u)A(\nabla u,\cdot)),P\phi\rangle$
(B.2)		$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\langle P(\Delta^{2}u)-P(A(\nabla u,\nabla u)D_{u}A(\nabla u,\nabla u))$
		$\displaystyle+2{\rm div}(A(\nabla u,\nabla u)A(\nabla u,P))-2A(\nabla u,\nabla u)A(\nabla u,\nabla P),\phi\rangle.$

Now let’s look at terms II and III which give the additional lower order terms to the intrinsic biharmonic map equation that make the conformal-harmonic map equation (1) conformally invariant.

	II	$\displaystyle=\frac{2}{3}\int_{\mathcal{M}}{\rm Sc}^{\mathcal{M}}\frac{d}{dt}\bigg{\|}_{t=0}\|du(t)\|^{2}=\frac{4}{3}\int_{\mathcal{M}}\langle{\rm Sc}^{\mathcal{M}}\nabla u,\nabla{(P\phi)}\rangle$
		$\displaystyle=\frac{4}{3}\int_{\mathcal{M}}\langle{\rm Sc}^{\mathcal{M}}\nabla_{e_{\alpha}}u,\nabla_{e_{\alpha}}(P\phi)\rangle=-\frac{4}{3}\int_{\mathcal{M}}\langle\nabla_{e_{\alpha}}({\rm Sc}^{\mathcal{M}}\nabla_{e_{\alpha}}u),P\phi\rangle$
		$\displaystyle=-\frac{4}{3}\int_{\mathcal{M}}\langle{\rm Sc}^{\mathcal{M}}\Delta u+\nabla{\rm Sc}^{\mathcal{M}}\nabla u,P\phi\rangle$
(B.3)			$\displaystyle=-\frac{4}{3}\int_{\mathcal{M}}\langle{\rm Sc}^{\mathcal{M}}(\Delta u+A(u)(\nabla{u},\nabla{u}))+\nabla{\rm Sc}^{\mathcal{M}}\nabla u,\phi\rangle.$

	III	$\displaystyle=-2\int_{\mathcal{M}}\frac{d}{dt}\bigg{\|}_{t=0}{\rm Ric}^{\mathcal{M}}(du(t),du(t))$
		$\displaystyle=-4\int_{\mathcal{M}}{\rm Ric}^{\mathcal{M}}(du,d(P\phi))$
		$\displaystyle=-4\int_{\mathcal{M}}\langle{{\rm Ric}^{\mathcal{M}}(du,\cdot),d(P\phi)}\rangle$
		$\displaystyle=4\int_{\mathcal{M}}\langle\nabla_{e_{\beta}}({\rm Ric}^{\mathcal{M}})^{\alpha\beta}\nabla_{e_{\alpha}}u+({\rm Ric}^{\mathcal{M}})^{\alpha\beta}\nabla^{2}_{\alpha\beta}{u},P\phi\rangle$
(B.4)			$\displaystyle=4\int_{\mathcal{M}}\langle\nabla_{\beta}({\rm Ric}^{\mathcal{M}})^{\alpha\beta}\nabla_{\alpha}u+({\rm Ric}^{\mathcal{M}})^{\alpha\beta}P(\nabla^{2}_{\alpha\beta}{u}),\phi\rangle,$

where we used the subscripts $\alpha,\beta$ to denote $e_{\alpha},e_{\beta}$ for short.

Combining these together, we get

	$\displaystyle\frac{d}{dt}\bigg{\|}_{t=0}{\mathcal{E}}(u(t),\mathcal{M})=$	$\displaystyle 2\int_{\mathcal{M}}\big{\langle}P(\Delta^{2}u)-P(A(\nabla u,\nabla u){D}_{u}A(\nabla u,\nabla u))$
		$\displaystyle+2{\rm div}(A(\nabla u,\nabla u)A(\nabla u,P))-2A(\nabla u,\nabla u)A(\nabla u,\nabla P)$
(B.5)			$\displaystyle-\frac{2}{3}{\rm Sc}^{\mathcal{M}}(\Delta u+A(u)(\nabla{u},\nabla{u}))-\frac{2}{3}\nabla{\rm Sc}^{\mathcal{M}}\nabla u$
		$\displaystyle+2\nabla_{\beta}({\rm Ric}^{\mathcal{M}})^{\alpha\beta}\nabla_{\alpha}u+2({\rm Ric}^{\mathcal{M}})^{\alpha\beta}P(\nabla^{2}_{\alpha\beta}{u}),\phi\big{\rangle}.$

Thus, the critical point $u$ of $\mathcal{E}$ satisfies the fourth order PDE:

	$\displaystyle P(\Delta^{2}u)=$	$\displaystyle P(A(\nabla u,\nabla u){D}_{u}A(\nabla u,\nabla u))-2{\rm div}(A(\nabla u,\nabla u)A(\nabla u,P))$
(B.6)			$\displaystyle+2A(\nabla u,\nabla u)A(\nabla u,\nabla P)+\frac{2}{3}{\rm Sc}^{\mathcal{M}}(\Delta u+A(u)(\nabla{u},\nabla{u}))$
		$\displaystyle+\frac{2}{3}\nabla{\rm Sc}^{\mathcal{M}}\nabla u-2\nabla_{\beta}({\rm Ric}^{\mathcal{M}})^{\alpha\beta}\nabla_{\alpha}u-2({\rm Ric}^{\mathcal{M}})^{\alpha\beta}P(\nabla^{2}_{\alpha\beta}{u}).$

Note that

	$\displaystyle P^{\bot}(\Delta^{2}u)$	$\displaystyle=\text{div}(P^{\bot}\nabla\Delta u)-\nabla P^{\bot}\nabla\Delta u$
		$\displaystyle=-\Delta A(\nabla u,\nabla u)-\Delta P^{\bot}\Delta u-2\nabla P^{\bot}\nabla\Delta u$
(B.7)			$\displaystyle=-\Delta A(\nabla u,\nabla u)+\Delta P\Delta u+2\nabla P\nabla\Delta u$
		$\displaystyle=-\Delta(A(\nabla u,\nabla u))-\Delta P\Delta u+2\text{div}(\nabla P\Delta u)\,,$

and

	$\displaystyle\Delta^{2}u$	$\displaystyle=P(\Delta^{2}u)+P^{\bot}(\Delta^{2}u)$
		$\displaystyle=P(\Delta^{2}u)-\Delta(A(\nabla u,\nabla u))-\Delta P\Delta u+2\text{div}(\nabla P\Delta u)\,.$

Therefore, (B) can be rewritten as

	$\displaystyle\Delta^{2}u=$	$\displaystyle-\Delta(A(\nabla u,\nabla u))-\Delta P\Delta u+2{\rm div}(\nabla P\Delta u)$
		$\displaystyle+P(A(\nabla u,\nabla u){D}_{u}A(\nabla u,\nabla u))-2{\rm div}(A(\nabla u,\nabla u)A(\nabla u,P))$
(B.8)			$\displaystyle+2A(\nabla u,\nabla u)A(\nabla u,\nabla P)+\frac{2}{3}{\rm Sc}^{\mathcal{M}}(\Delta u+A(u)(\nabla{u},\nabla{u}))$
		$\displaystyle+\frac{2}{3}\nabla{\rm Sc}^{\mathcal{M}}\nabla u-2\nabla\text{Ric}^{\mathcal{M}}(\nabla u,\cdot)-2{\rm Ric}^{\mathcal{M}}(P(\nabla^{2}u),\cdot)$

or equivalently (c.f. [32, equation (1.2)], [31, equation (9)])

	$\displaystyle\Delta^{2}u=$	$\displaystyle-\Delta(\nabla P^{\bot}\nabla u)-{\rm div}(\nabla P^{\bot}\Delta u)+2\nabla P^{\bot}\nabla(\nabla P^{\bot}\nabla u)$
		$\displaystyle+2\nabla P^{\bot}\nabla P^{\bot}\Delta u-(\nabla PP^{\bot}-P^{\bot}\nabla P)\nabla\Delta u$
(B.9)			$\displaystyle+P\langle\nabla P^{\bot}\nabla u,{D}_{u}({D}_{u}P^{\bot})\nabla u\nabla u\rangle-2{\rm div}\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}P\rangle$
		$\displaystyle+2\langle\nabla P^{\bot}\nabla u,\nabla P^{\bot}\nabla P\rangle+\frac{2}{3}{\rm Sc}^{\mathcal{M}}(\Delta u+\nabla P^{\perp}\nabla u)$
		$\displaystyle+\frac{2}{3}\nabla{\rm Sc}^{\mathcal{M}}\nabla u-2\nabla\text{Ric}^{\mathcal{M}}(\nabla u,\cdot)-2{\rm Ric}^{\mathcal{M}}(P(\nabla^{2}u),\cdot)\,,$

where

\nabla\text{Ric}^{\mathcal{M}}(\nabla u,\cdot):=\nabla_{\beta}({\rm Ric}^{\mathcal{M}})^{\alpha\beta}\nabla_{\alpha}u

and

{\rm Ric}^{\mathcal{M}}(P(\nabla^{2}u),\cdot):=({\rm Ric}^{\mathcal{M}})^{\alpha\beta}P(\nabla^{2}_{\alpha\beta}{u})\,.

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	$\displaystyle\int_{\mathcal{M^{\epsilon}}}\frac{\|w\|^{2}}{\rho^{4}}dv_{g}$	$\displaystyle\leq\frac{L^{4}}{\delta^{2}}\int_{\mathcal{M}^{2\epsilon}}\frac{\|\tilde{w}\|^{2}\|\nabla\psi\|^{2}}{\psi^{4}}dv_{g}$
		$\displaystyle\leq\frac{C(\mathcal{M})(1+\delta^{-2})^{2}L^{4}}{\delta^{2}}\int_{\mathcal{M}^{2\epsilon}}\|\Delta\tilde{w}\|^{2}dv_{g}$
(1.7)			$\displaystyle\leq\frac{C(\mathcal{M},\epsilon)(1+\delta^{-2})^{2}L^{4}}{\delta^{2}}\int_{\mathcal{M}}\|\Delta w\|^{2}dv_{g}\,.$

	$\displaystyle\int_{\mathcal{M_{\epsilon}}}\frac{\|w\|^{2}}{\rho^{4}}dv_{g}$	$\displaystyle\leq m_{\epsilon}^{-\frac{1}{2}}\epsilon^{-4}\int_{\mathcal{M}}\|w\|^{2}\psi^{\frac{1}{2}}dv_{g}$
		$\displaystyle\leq 8m_{\epsilon}^{-\frac{1}{2}}\epsilon^{-4}\lambda_{1}^{-1}D^{\frac{1}{2}}\int_{\mathcal{M}}\|\nabla w\|^{2}dv_{g}$
(1.8)			$\displaystyle\leq C(\mathcal{M},\epsilon,m_{\epsilon},\lambda_{1},D)\int_{\mathcal{M}}\|\Delta w\|^{2}dv_{g}\,.$

	$\displaystyle\int_{2U_{i}}\|\bar{\Delta}\bar{u}_{i}\|^{2}dx$	$\displaystyle=\int_{B_{2r_{i}}(p_{i})}\|\bar{\Delta}u\|^{2}dv_{\bar{g}}$
(4.7)			$\displaystyle\leq C_{i}\int_{B_{2r_{i}}(p_{i})}(\|\Delta u\|^{2}+\|\nabla u\|^{2})dv_{g}\leq C_{i}\sqrt{\epsilon_{0}}$

		$\displaystyle\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}=\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)-v\|\bar{\nabla}v\|^{2}+u\|\bar{\nabla}u\|^{2}\|^{2}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+2\int_{\mathcal{M}}\left\|v\|\bar{\nabla}v\|^{2}-u\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
	$\displaystyle=$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+2\int_{\mathcal{M}}\left\|v\left(\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2}\right)+(v-u)\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\left\|\|\bar{\nabla}v\|^{2}-\|\bar{\nabla}u\|^{2}\right\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+4\left(\int_{\mathcal{M}}\|\bar{\nabla}(v+u)\|^{4}dv_{\bar{g}}\right)^{\frac{1}{2}}\left(\int_{\mathcal{M}}\|\bar{\nabla}(v-u)\|^{4}dv_{\bar{g}}\right)^{\frac{1}{2}}dv_{\bar{g}}$
		$\displaystyle+4\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}$
	$\displaystyle\leq$	$\displaystyle 2\int_{\mathcal{M}}\|\bar{\tau}(v)-\bar{\tau}(u)\|^{2}dv_{\bar{g}}+C\sqrt{\epsilon_{0}}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}+4\int_{\mathcal{M}}\|v-u\|^{2}\|\bar{\nabla}u\|^{4}dv_{\bar{g}}\,,$

	$\displaystyle 2\int_{\mathcal{M}}\left\langle P^{\bot}(\bar{\Delta}^{2}u),v-u\right\rangle dv_{\bar{g}}$	$\displaystyle\geq-C\int_{\mathcal{M}}\left\|(v-u)^{\bot}\right\|\cdot\left\|P^{\bot}(\Delta^{2}u)\right\|dv_{\bar{g}}$
		$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|v-u\|^{2}\rho(x)^{-4}dv_{g}$
		$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\Delta(v-u)\|^{2}dv_{g}$
(4.21)			$\displaystyle\geq-C{\epsilon_{0}}^{1/4}\int_{\mathcal{M}}\|\bar{\Delta}(v-u)\|^{2}dv_{\bar{g}}\,.$

Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Theorem 1.2.

Remark 1.3.

Theorem 1.4.

Theorem 1.5.

Proof.

Corollary 1.6.

Acknowledgement

2. Preliminary

Remark 2.1.

Theorem 2.2.

3. Second-order Hardy inequality on manifolds

Theorem 3.1.

Theorem 3.2.

Proof.

4. Proof of the main result

Remark 4.1.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Proof of Theorem 1.2.

5. Conflict of interest and Data availability

Appendix A Conformal invariance of ℰ​(u,ℳ)\mathcal{E}(u,\mathcal{M})

Proposition A.1.

Appendix B Conformal-harmonic map equation

References

Appendix A Conformal invariance of $\mathcal{E}(u,\mathcal{M})$