Analytically stable Higgs bundles on some non-K\"ahler manifolds

TL;DR

This paper investigates the existence of Hermitian-Einstein metrics on analytically stable Higgs bundles over certain non-Kähler Hermitian manifolds, expanding the understanding of Higgs bundle stability beyond Kähler geometry.

Contribution

It provides conditions under which Hermitian-Einstein equations can be solved for stable Higgs bundles on non-compact, non-Kähler Hermitian manifolds, a setting less explored in the literature.

Findings

Existence of Hermitian-Einstein metrics on stable Higgs bundles established

Conditions identified for solving the Hermitian-Einstein equation in non-Kähler settings

Extension of stability results to non-compact Hermitian manifolds

Abstract

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily K\"ahler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.

Equations

$$\sqrt{-1}{\Lambda }_{\omega }(F_H-\frac{1}{r}\text{tr}{F}_{H}I{d}_E)=0,$$

$$F_{H,\theta }=F_H+[\theta ,{\theta }^{{\ast }_H}]+{\partial }_H\theta +{\bar{\partial }}_E{\theta }^{{\ast }_H},$$

$$\sqrt{-1}{\Lambda }_{\omega }{F}_{H,\theta }^{\perp }=0.$$

$$\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }f\ge -B\phi$$

$$\underset{x\in M}{\sup }f(x)\le C_{1}B+C_2{\int }_{M}f\phi \frac{{\omega }^n}{n!}.$$

$$|\sqrt{-1}{\Lambda }_{\omega }{F}_{H_0,\theta }{|}_{H_0}\le \hat{B}\phi$$

$${\text{deg}}_{\omega }(E,H_0)=\sqrt{-1}{\int }_M\text{tr}({\Lambda }_{\omega }{F}_{H_0,\theta })\frac{{\omega }^n}{n!}.$$

$${\text{deg}}_{\omega }(\mathcal{S},H_0)={\int }_{M\setminus {\Sigma }_{\mathcal{S}}}(\sqrt{-1}\text{tr}({\pi }_{\mathcal{S}}{\Lambda }_{\omega }{F}_{H_0,\theta })-|{\overline{\partial }}_{\theta }{\pi }_{\mathcal{S}}{|}_{H_0}^2)\frac{{\omega }^n}{n!},$$

$$\frac{{\text{deg}}_{\omega }(\mathcal{S},H_0)}{\text{rank}(\mathcal{S})}<(\le )\frac{{\text{deg}}_{\omega }(E,H_0)}{\text{rank}(E)}.$$

$$\{\begin{array}{rl}& \sqrt{-1}{\Lambda }_{\omega }{F}_{H_i}^{\perp }=0,\\ & H_i{|}_{\partial M_i}=H_0.\end{array}$$

$${\int }_M\mathrm{tr}(\Phi (H_0,\theta )s)\frac{{\omega }^n}{n!}+{\int }_M⟨\Psi (s)({\overline{\partial }}_{\theta }s),{\overline{\partial }}_{\theta }s{⟩}_{H_0}\frac{{\omega }^n}{n!}={\int }_M\mathrm{tr}(\Phi (H,\theta )s)\frac{{\omega }^n}{n!},$$

$$\Phi (H,\theta )=\sqrt{-1}{\Lambda }_{\omega }(F_H+[\theta ,{\theta }^{{\ast }_H}])$$

$$\Psi(x,y)=\left\{\begin{split}&\frac{e^{y-x}-1}{y-x},\ \ \ &x\neq y;\\ &\ \ \ \ 1,\ \ \ \ \ \ &x=y.\end{split}\right.$$

$${\mathcal{S}}_{H_0}(E)=\{\eta \in {\Omega }^0(M,End(E))|{\eta }^{\ast H_0}=\eta \},$$

$$h=H_0^{-1}H=\exp s,$$

$$\begin{array}{rl}& \log (\frac{1}{2r}(\text{tr}h+\text{tr}{h}^{-1}))\le |s{|}_{H_0}\le r^{\frac{1}{2}}\log (\text{tr}h+\text{tr}{h}^{-1});\\ & {\partial }_H-{\partial }_{H_0}=h^{-1}{\partial }_{H_0}h;\\ & F_H-F_{H_0}={\overline{\partial }}_E(h^{-1}{\partial }_{H_0}h);\\ & {\theta }^{\ast H}=h^{-1}{\theta }^{\ast H_0}h,\end{array}$$

$$\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }\log (\text{tr}h)\ge -|\Phi (H_0,\theta ){|}_{H_0}-|\Phi (H,\theta ){|}_H$$

$$\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }\log (\text{tr}h+\text{tr}{h}^{-1})\ge -|\Phi (H_0,\theta ){|}_{H_0}-|\Phi (H,\theta ){|}_H.$$

$$\{\begin{array}{rl}& \sqrt{-1}{\Lambda }_{\omega }(F_H+[\theta ,{\theta }^{{\ast }_H}])=\lambda I{d}_E,\\ & H{|}_{\partial X}=H_0,\end{array}$$

$$\sqrt{-1}{\Lambda }_{\omega }(F_H+[\theta ,{\theta }^{{\ast }_H}])=(\lambda -\frac{\sqrt{-1}}{r}{\Lambda }_{\omega }\partial \overline{\partial }f)I{d}_E.$$

$$\{\begin{array}{rl}& \sqrt{-1}{\Lambda }_{\omega }(F_H+[\theta ,{\theta }^{{\ast }_H}]{)}^{\perp }=0,\\ & H{|}_{\partial X}=H_0.\end{array}$$

$$\eta =\sum _{\alpha =1}^r{\lambda }_{\alpha }\cdot e_{\alpha }\otimes e^{\alpha },$$

$$\Psi (\eta )(A)=\Psi ({\lambda }_{\beta },{\lambda }_{\alpha })A_{\beta }^{\alpha }{e}_{\alpha }\otimes e^{\beta }.$$

$${\int }_{\overline{X}}\mathrm{tr}(\Phi (H_0,\theta )s)\frac{{\omega }^n}{n!}+{\int }_{\overline{X}}⟨\Psi (s)({\overline{\partial }}_{\theta }s),{\overline{\partial }}_{\theta }s{⟩}_{H_0}\frac{{\omega }^n}{n!}={\int }_{\overline{X}}\mathrm{tr}(\Phi (H,\theta )s)\frac{{\omega }^n}{n!},$$

$$\frac{1}{r}\text{tr}(\exp s)\ge (\det h{)}^{\frac{1}{r}}=1$$

$${\tilde{\Delta }}_{\omega }f=2\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }f.$$

$$({\tilde{\Delta }}_{\omega }-{\Delta }_g)f=⟨V,\nabla f{⟩}_g,$$

$$\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }f\ge \eta$$

$${\int }_{M}f\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }(\psi \frac{{\omega }^{n-1}}{(n-1)!})\ge {\int }_M\eta \psi \frac{{\omega }^n}{n!}.$$

$$\sqrt{-1}{\Lambda }_{\omega }\partial \overline{\partial }\log (\text{tr}(\exp s)/r{)}^{\sim }\ge -|{\Lambda }_{\omega }{F}_{H_0,\theta }^{\perp }{|}_{H_0}^{\sim X}$$

Full text

*2010 Mathematics Subject Classification: 53C07; 14J60; 32Q15.

Key words and phrases: Higgs bundles; Gauduchon manifold; Hermitian-Einstein equation; non-compact*

1. Introduction

Let $(M,\omega)$ be an $n$-dimensional Hermitian manifold and $(E,\overline{\partial}_{E})$ a $r$-rank holomorphic vector bundle on $M$. A Hermitian metric $H$ on the bundle $E$ is called $\omega$-Hermitian-Einstein if it satisfies the following Hermitian-Einstein equation on $M$, i.e.

[TABLE]

where $F_{H}$ is the curvature tensor of Chern connection $D_{H}$ with respect to $H$ and $\Lambda_{\omega}$ denotes the contraction with the Hermitian metric $\omega$.

When $(M,\omega)$ is a compact Kähler manifold, by the famous Donaldson-Uhlenbeck-Yau theorem ([28, 9, 31]), we know the holomorphic vector bundle $(E,\overline{\partial}_{E})$ must have an $\omega$-Hermitian-Einstein metric if $(E,\overline{\partial}_{E})$ is $\omega$-stable in the sense of Mumford-Takemoto. This classical result has a lot of interesting and important generalizations and extensions (see [18, 13, 29, 5, 6, 11, 14, 1, 2, 7, 15, 16, 17, 26, 22, 23], etc.).

A Higgs bundle $(E,\overline{\partial}_{E},\theta)$ over $M$ is a holomorphic bundle $(E,\overline{\partial}_{E})$ coupled with a Higgs field $\theta\in\Omega_{X}^{1,0}(\mathrm{End}(E))$ such that $\overline{\partial}_{E}\theta=0$ and $\theta\wedge\theta=0$. Higgs bundles first emerged thirty years ago in Hitchin’s ([13]) reduction of self-dual equation on $R^{4}$ to Riemann surface and in Simpson’s ([29]) work on nonabelian Hodge theory, they have rich structures and play an important role in many different areas including gauge theory, Kähler and hyperkähler geometry, group representations and nonabelian Hodge theory. Letting $H$ be a Hermitian metric on the bundle $E$, we consider the Hitchin-Simpson connection: $D_{H,\theta}=D_{H}+\theta+\theta^{*_{H}}$, where $\theta^{*_{H}}$ is the adjoint of $\theta$ with respect to the metric $H$. The curvature of this connection is

[TABLE]

where $\partial_{H}$ is the $(1,0)$-part of $D_{H}$. A Hermitian metric $H$ is said to be a Hermitian-Einstein metric on Higgs bundle $(E,\overline{\partial}_{E},\theta)$ if it satisfies

[TABLE]

where $F_{H,\theta}^{\bot}$ is the trace-free part of the curvature of the Hitchin-Simpson connection.

The Donaldson-Uhlenbeck-Yau theorem was generalized to the Higgs bundles case by Hitchin ([13]) and Simpson ([29], [30]). Simpson ([29]) even studied some non-compact Kähler manifolds case. Under some assumption for the base manifold, he proved that the analytic stability implies the existence of Hermitian-Einstein metric. The Donaldson-Uhlenbeck-Yau theorem for the non-compact base manifold case is important and interesting ([16, 22, 23, 24]). Recently, Mochizuki ([25]) made an important progress in this direction. He weakened the assumption in Simpson’s result such that the volume of base manifold may not be finite and he also studied the curvature decay of the Hermitian-Einstein metrics.

In this paper, we study the non-Kähler case. A Hermitian metric $\omega$ is called to be Gauduchon if it satisfies $\partial\overline{\partial}\omega^{n-1}=0$. If $M$ is compact, it has been proved by Gauduchon ([12]) that there exists a Gauduchon metric in the conformal class of every Hermitian metric $\omega$. When the base Hermitian manifold is compact and Gauduchon, the Donaldson-Uhlenbeck-Yau theorem is also valid (see [3, 4, 8, 18, 20, 21]). Inspired by Mochizuki’s result ([25]), we consider the case that the base manifold $(M,\omega)$ is non-compact Gauduchon and satisfies the following assumption.

Assumption 1. Let $\varphi$ be a nonnegative function on $(M,\omega)$ with $\int_{M}\varphi\frac{\omega^{n}}{n!}<+\infty$. There exist positive constants $C_{i}$ (i=1, 2) such that for any nonnegative bounded function $f$ satisfying

[TABLE]

in weakly sense (see definition (2.4) for details) for a positive number $B$, we have

[TABLE]

Moreover, if the function $f$ satisfies $\sqrt{-1}\Lambda_{\omega}\partial\overline{\partial}f\geq 0$ on $M$, then we have $\sqrt{-1}\Lambda_{\omega}\partial\overline{\partial}f\equiv 0$.

Let the background metric $H_{0}$ be a Hermitian metric of $E$ such that

[TABLE]

for some constant $\hat{B}>0$. Define the analytic degree of $E$ to be the real number

[TABLE]

As in [29], we define the analytic degree of any saturated sub-Higgs sheaf $\mathcal{S}$ of $(E,\overline{\partial}_{E},\theta)$ by

[TABLE]

where $\Sigma_{\mathcal{S}}$ denotes the set of singularities where $\mathcal{S}$ is not locally free, $\overline{\partial}_{\theta}:=\overline{\partial}_{E}+\theta$ and $\pi_{\mathcal{S}}$ denotes the projection onto $\mathcal{S}$ with respect to the metric $H_{0}$ outside $\Sigma_{\mathcal{S}}$. When the base Gauduchon manifold $(M,\omega)$ is compact, it is easy to see that the analytic degree $\textmd{deg}_{\omega}(\mathcal{S},H_{0})$ is independent of the choice of the background metric $H_{0}$.

Following [29], we say that the Higgs bundle $(E,\overline{\partial}_{E},\theta)$ is $H_{0}$-analytic stable (semi-stable) if for every proper saturated sub-Higgs sheaf $\mathcal{S}\subset E$, it holds

[TABLE]

Now we give our main theorem as follows.

Theorem 1.1**.**

Let $(M,\omega)$ be a non-compact Gauduchon manifold satisfying the assumption 1, $(E,\overline{\partial}_{E},\theta)$ a Higgs bundle over $M$ and $H_{0}$ a background Hermitian metric on $E$ satisfying the condition (1.5). If $(E,\overline{\partial}_{E},\theta)$ is analytically $H_{0}$-stable, then there exists a Hermitian-Einstein metric $H$ satisfying the below conditions:

1. (i)

$\det(H)=\det(H_{0})$. 2. (ii)

Set $h=H_{0}^{-1}H$. Then, $|h|_{H_{0}}$ and $|h^{-1}|_{H_{0}}$ are bounded, and $\int_{M}(|\overline{\partial}_{E}h|_{H_{0}}^{2}+|[\theta,h]|_{H_{0}}^{2})\frac{\omega^{n}}{n!}<+\infty$.

The above theorem can be seen as a generalization of Mochizuki’s result ([25]) to the non-Kähler case. In [25], Mochizuki proved the existence of an exhaustion function $\phi$ on $M$. Fix a number $a_{i}$ and let $M_{i}$ denote the compact space $\phi(x)\leq a_{i}$ with boundary $\partial M_{i}$, so we can take a sequence of exhaustion compact subsets $M_{i}$ in $M$ with $\cup M_{i}=M$. Let’s consider the Dirichlet problem on $M_{i}$:

[TABLE]

According to the results of Donaldson ([10], [33] for the Hermitian manifold case), we know that there exists a unique Hermitian metric $H_{i}$ satisfying the above Dirichlet problem (1.8) and $\det(H_{i})=\det(H_{0})$ on $M_{i}$. Following the idea in [25], one can take the limit as $a_{i}\rightarrow+\infty$ and get the convergence $H_{\infty}$ of a subsequence of $H_{i}$ on any compact subset, which should satisfy $\sqrt{-1}\Lambda_{\omega}F_{H_{\infty}}^{\bot}=0$ on the whole $M$. Now the key is to obtain a $C^{0}$-bound. When the base manifold is Kähler, Mochizuki ([25]) introduced the Donaldson’s functional on the space of Hermitian metrics satisfying the Dirichlet boundary condition. The Donaldson’s functional played a key role in Mochizuki’s proof of the uniform $C^{0}$-bound. However, in the non-Kähler case, the Donaldson functional may not be well-defined. So we need new argument in our case. In fact, our argument relies on the following identity:

[TABLE]

where $s=\log(H_{0}^{-1}H)$,

[TABLE]

and

[TABLE]

The above identity (1.9) was proved in [27] for the closed Gauduchon manifold case, and in [32] for the compact Gauduchon manifold with non-empty boundary and some non-compact case.

This paper is organized as follows. In Section 2, we give some estimates and preliminaries for the Hermitian-Einstein equation (1.2). In Section 3, we give a proof of Theorem 1.1 by using the identity (1.9). In Section 4, we study the uniqueness of Hermitian-Einstein metric in Theorem 1.1.

Acknowledgement: The two authors are partially supported by NSF in China No.11625106, 11571332 and 11721101. The first author is also supported by NSF in China No.11801535, the China Postdoctoral Science Foundation (No.2018M642515) and the Fundamental Research Funds for the Central Universities.

2. Preliminary results

Let $(M,\omega)$ be a Hermitian manifold and $(E,\overline{\partial}_{E},\theta)$ a Higgs bundle over $M$. Letting $H_{0}$ and $H$ be two Hermitian metrics on the bundle $E$, we denote

[TABLE]

and set

[TABLE]

where $s\in\mathcal{S}_{H_{0}}(E)\cap\mathcal{S}_{H}(E)$. It is easy to check the following identities

[TABLE]

where $r=rank(E)$ and $\partial_{H_{0}}$ is the $(1,0)$ part of the Chern connection $D_{H_{0}}$. Furthermore, we have the following estimates (Lemma 3 (d) in [29], Proposition 2.7 in [33])

[TABLE]

and

[TABLE]

The Dirichlet problem for the Hermitian-Einstein equation was first solved in [10] by Donaldson for the Kähler manifold case, in [33] for the general Hermitian manifold case. The following proposition was proved in [32].

Proposition 2.1** (Theorem 4.1 in [33]).**

Let $(E,\bar{\partial}_{E},\theta)$ be a Higgs bundle over a compact Hermitian manifold $(\overline{X},\omega)$ with non-empty boundary $\partial X$ and $H_{0}$ a Hermitian metric on $E$. There is a unique Hermitian metric $H$ on $E$ such that

[TABLE]

where $\lambda$ is a constant.

Let $\tilde{H}$ be a solution of (2.6), $f=\log\frac{\det(H_{0})}{\det(\tilde{H})}$ and $H:=e^{\frac{f}{r}}\tilde{H}$. It is easy to check that

[TABLE]

So we have:

[TABLE]

Conversely, if we have a solution of (2.8), then we can get a solution of (2.6) by conformal transformation. Using the uniqueness of (2.6), one can easily prove the uniqueness of (2.8). So we obtain the following corollary.

Corollary 2.2**.**

Let $(E,\bar{\partial}_{E},\theta)$ be a Higgs bundle over a compact Hermitian manifold $(\overline{X},\omega)$ with non-empty boundary $\partial X$ and $H_{0}$ a Hermitian metric on $E$. There is a unique Hermitian metric $H$ on $E$ with $det(H_{0}^{-1}H)=1$ on $\overline{X}$ satisfying (2.8).

Given $\eta\in\mathcal{S}_{H_{0}}(E)$, we can choose a local unitary basis $\{e_{\alpha}\}_{\alpha=1}^{r}$ with respect to $H_{0}$ and local functions $\{\lambda_{\alpha}\}_{\alpha=1}^{r}$ such that

[TABLE]

where $\{e^{\alpha}\}_{\alpha=1}^{r}$ denotes the dual basis in $E^{*}$. Let $\Psi\in C^{\infty}(\mathbb{R}\times\mathbb{R},\mathbb{R})$ and $A=\sum\limits^{r}_{\alpha,\beta=1}A_{\beta}^{\alpha}e_{\alpha}\otimes e^{\beta}\in\text{End}(E)$. We define:

[TABLE]

Proposition 2.3** (Proposition 2.6 in [32]).**

Let $(E,\bar{\partial}_{E},\theta)$ be a Higgs bundle with a fixed Hermitian metric $H_{0}$ over a compact Gauduchon manifold $(\overline{X},\omega)$ with non-empty smooth boundary $\partial X$. Let $H$ be a Hermitian metric on $E$ satisfying $H|_{\partial X}=H_{0}|_{\partial X}$. Then we have the following identity:

[TABLE]

where $s:=\log(H^{-1}_{0}H)$ and $\Psi$ is the function which is defined in (1.11).

In the following, we always assume that the compact manifold $\overline{X}$ with non-empty smooth boundary $\partial X$ is a subset of the Hermitian manifold $(M,\omega)$. Let $H$ be the unique Hermitian metric in Corollary 2.2. Set $\exp{s}=h=H_{0}^{-1}H$. By the condition $\det(h)\equiv 1$ on $\overline{X}$ and the relationship between the geometric mean and arithmetic mean, one can get that

[TABLE]

on $\overline{X}$. As that in [25], we extend $\log(\text{tr}(\exp s)/r)$ and $(|\Lambda_{\omega}F_{H_{0},\theta}|_{H_{0}})|_{\overline{X}}$ to the functions $\log(\text{tr}(\exp s)/r)^{\thicksim}$ and $|\Lambda_{\omega}F_{H_{0},\theta}|_{H_{0}}^{\thicksim X}$ on the whole $M$ by setting [math] outside $\overline{X}$.

Let $g$ be the Hermitian metric with respect to $\omega$. As usual, we denote the Beltrami-Laplace operator on the Hermitian manifold $(M,\omega)$ by $\Delta_{g}$, and define the complex Laplace operator $\widetilde{\Delta}_{\omega}$ for functions as

[TABLE]

It is well known that the difference of the two Laplacians is given by a first order differential operator as follows

[TABLE]

where $V$ is a smooth vector field on $M$. Usually the complex Laplace operator is not a self adjoint operator.

Definition 2.4**.**

A function $f$ on the Hermitian manifold $(M,\omega)$ satisfying

[TABLE]

in weakly sense means that, for any nonnegative compactly supported smooth function $\psi$, there holds

[TABLE]

Proposition 2.5**.**

We have

[TABLE]

in weak sense on $M$.

Proof Due to the inequality (2.10) and the boundary condition of $H$, we know

[TABLE]

where $\nu$ is the outer normal vector field at $\partial{X}$. Direct computations give us that

[TABLE]

where $\psi$ is a test function. This means that

[TABLE]

Similarly, we have

[TABLE]

Combining the above two equalities, we deduce that

[TABLE]

Now we should use the following result.

Lemma 2.6**.**

Let $(M,\omega)$ be a Hermitian manifold, $g$ be the Riemannian metric with respect to $\omega$. Then we have

[TABLE]

where $\ast$ is the Hodge star operator with respect to the Riemannian metric $g$ and $f$ is a differential function on $M$.

Proof Let $\theta$ be a $1$-form. By a simple calculation, we obtain that

[TABLE]

and

[TABLE]

Therefore, because of the arbitrary of $\theta$, the above lemma follows.

Note that

[TABLE]

One can easily check that

[TABLE]

Hence we derive

[TABLE]

This implies

[TABLE]

This conclude the proof of the proposition.

$\Box$

3. A proof of Theorem 1.1

Let’s first recall the following lemmas which are proved by Mochizuki in [25].

Lemma 3.1**.**

(Lemma 2.20 in [25]) There exists an exhaustion function $\phi\in C^{\infty}(M)$.

Lemma 3.2**.**

(Lemma 2.21 in [25]) There is a sequence of compact subsets $M_{i}\subset M_{i+1}\subset M(i=1,2,\ldots)$ with $\cup M_{i}=M$, where each $M_{i}$ is a submanifold with non-empty smooth boundary $\partial M_{i}$ such that $M_{i}\setminus\partial M_{i}$ is an open subset of $M$. Moreover, each connected component of $M_{i}$ has non-empty boundary.

Fix a number $a_{i}$ and let $M_{i}$ denote the compact space $\{x\in M|\phi(x)\leq a_{i}\}$ with boundary $\partial M_{i}$. By choosing a sequence $a_{i}\rightarrow+\infty$ such that each $a_{i}$ is not a critical value of $\phi$, we have a sequence of exhaustion compact subsets $M_{i}\subset M$ with non-empty smooth boundary. Let’s consider the Dirichlet problem on $M_{i}$. Then corollary 2.2 tells us that there exists a Hermitian metric $H_{i}$ on $E|_{M_{i}}$ such that

[TABLE]

Set $H_{0}^{-1}H_{i}=h_{i}=\exp s_{i}$. We already know on $M_{i}$, it holds that

[TABLE]

Now extend $\log(\text{tr}(\exp s_{i})/r)$ and $(|\Lambda_{\omega}F_{H_{0},\theta}^{\bot}|_{H_{0}})|_{M_{i}}$ to the functions $\log(\text{tr}(\exp s_{i})/r)^{\thicksim}$ and $|\Lambda_{\omega}F_{H_{0},\theta}^{\bot}|_{H_{0}}^{\thicksim M_{i}}$ on $M$ by setting [math] outside $M_{i}$. According to Proposition 2.5, we obtain

[TABLE]

in weak sense on $M$. From the assumptions in Theorem 1.1, it can be seen that there exists two positive constants $C_{1}$, $C_{2}$ such that for any $i$, we have

[TABLE]

Note that $\text{tr}s_{i}=0$, it is easy to check that

[TABLE]

So clearly it implies that

[TABLE]

where $C_{3}$ and $C_{4}$ are positive constants depending only on $C_{1}$, $C_{2}$, $\hat{B}$ and $r$.

Proof of Theorem 1.1

When the Higgs bundle $(E,\bar{\partial}_{E},\theta)$ is $H_{0}$-analytically stable on $(M,\omega)$, we will show that, by choosing a subsequence, $H_{i}$ converge to a Hermitian-Einstein metric $H_{\infty}$ in $C_{loc}^{\infty}$-topology as $i\rightarrow+\infty$.

(1) Uniform $C^{0}$-estimate. By (3.6), the key is to get a uniform estimate for $\int_{M_{i}}|s_{i}|_{H_{0}}\cdot\varphi\frac{\omega^{n}}{n!}$, i.e. there exists a constant $\hat{C}$ independent of $i$, such that

[TABLE]

for all $i$.

As that in [25], we prove (3.7) by contradiction. If not, there would exist a subsequence $i\rightarrow+\infty$ such that $l_{i}\rightarrow+\infty$. Set

[TABLE]

Then, we have

[TABLE]

and

[TABLE]

Now, we show that $\|u_{i}\|_{L^{2}_{1}}$ are uniformly bounded on any compact subset of $M$.

Based on Proposition 2.3, we deduce

[TABLE]

By the definition (1.11), it is easy to check that

[TABLE]

increases monotonically as $l\rightarrow+\infty$. Let $\varsigma\in C^{\infty}(\mathbb{R}\times\mathbb{R},\mathbb{R}^{+})$ satisfying $\varsigma(x,y)<(x-y)^{-1}$ whenever $x>y$. Clearly Eqs. (3.11), (3.12) and the arguments in [29, Lemma 5.4] yield that

[TABLE]

From (3.10), we may assume that $(x,y)\in(-C_{5},C_{5})\times(-C_{5},C_{5})$. Note that $\frac{1}{2C_{5}}<\frac{1}{x-y}$ when $x>y$. In particular, taking $\zeta(x,y)=\frac{1}{2C_{5}}$ in (3.13), we immediately get

[TABLE]

for $i\gg 0$, and then

[TABLE]

where $C_{6}$ is a uniform constant. Thus, $u_{i}$ are bounded in $L_{1}^{2}$ on any compact subset $M$. By choosing a subsequence, we have $u_{i}\rightharpoonup u_{\infty}$ weakly in $L^{2}_{1,loc}$. Of course $\text{tr}s_{i}=0$ and (3.10) imply that

[TABLE]

The condition $\int_{M}\varphi\frac{\omega^{n}}{n!}<+\infty$ means that, for any $\epsilon>0$, there exists $i_{0}$ such that

[TABLE]

for all $i\geq i_{0}$. Combining this and (3.9), (3.10) gives that

[TABLE]

for any $i_{0}\leq i\leq j$. Noting that $L_{1}^{2}\hookrightarrow L^{1}$ on any compact subset, we have

[TABLE]

for all $i_{0}\leq i$, and then

[TABLE]

This indicates that

[TABLE]

and $u_{\infty}$ is non-trivial. If $i_{0}\leq i\leq j$, we derive

[TABLE]

where we have used (3.17), (3.13) and (1.5). Taking limits $j\rightarrow\infty$ and $i\rightarrow\infty$, one can obtain

[TABLE]

The fact that $\epsilon$ is arbitrary in the above inequality obviously implies

[TABLE]

Now following Simpson’s argument [29, Lemma 5.5], we conclude that the eigenvalues of $u_{\infty}$ are constant almost everywhere. Let $\mu_{1}<\mu_{2}<\cdots<\mu_{l}$ be the distinct eigenvalues of $u_{\infty}$. Because $\textmd{tr}(u_{\infty})=0$ and $u_{\infty}\neq 0$, there must hold that $2\leq l\leq r$. For each $\mu_{\alpha}(1\leq\alpha\leq l-1)$, we construct a function $P_{\alpha}:\mathbb{R}\rightarrow\mathbb{R}$ such that

[TABLE]

Setting $\pi_{\alpha}=P_{\alpha}(u_{\infty})$, from [29, p.887], we have: (i) $\pi_{\alpha}\in L^{2}_{1}$; (ii)$\pi_{\alpha}^{2}=\pi_{\alpha}=\pi_{\alpha}^{*_{H_{0}}}$; (iii) $(\textmd{Id}_{E}-\pi_{\alpha})\bar{\partial}\pi_{\alpha}=0$ and (iv) $(\textmd{Id}_{E}-\pi_{\alpha})[\theta,\pi_{\alpha}]=0$. By Uhlenbeck and Yau’s regularity statement of $L^{2}_{1}$-subbundle [31], $\{\pi_{\alpha}\}_{\alpha=1}^{l-1}$ determine $l-1$ Higgs sub-sheaves $\{\mathcal{E}_{\alpha}\}_{\alpha=1}^{l-1}$ of $E$. Using the same argument as that ([29, Proposition 5.3]), we can prove that there must exist a Higgs subsheaf $\mathcal{E}_{\alpha}$ which contradicts the stability of $(E,\bar{\partial}_{E},\theta)$. This completes the proof of uniform $C^{0}$-estimate.

(2) Uniform local $C^{1}$-estimate. From the property that $H_{i}$ satisfies the Hermitian-Einstein equation (1.2) and $\det{h_{i}}=1$ on $M_{i}$, it is easy to see that

[TABLE]

[TABLE]

and then

[TABLE]

Let $T_{i}=h_{i}^{-1}\partial_{H_{0}}h_{i}$. A direct computation gives us that

[TABLE]

where the constants $C_{7},C_{8},C_{9}$ depend only on the dimension $n$ and the rank $r$. We will follow the argument in [19, Lemma 2.4] to get local uniform $C^{1}$-estimate. Let $\Omega$ be a compact subset in $M$, $d$ be a constant less than the distance of $\Omega$ to $\partial M_{i_{0}}$, where $i_{0}$ is large enough such that $\Omega\subset M_{i_{0}}$. Set $\Omega_{1}=\{x\in M|\mathrm{dist}(x,\Omega)\leq\frac{1}{4}d\}$ and $\Omega_{2}=\{x\in M|\mathrm{dist}(x,\Omega)\leq\frac{1}{2}d\}$. Let’s choose two non-negative cut-off functions $\psi_{1}$, $\psi_{2}$ such that:

[TABLE]

[TABLE]

and

[TABLE]

where $C_{10}$ is a constant depending only on $d^{-2}$ and the geometry of $(\Omega_{2},\omega)$. Consider the following test function

[TABLE]

where the constant $\tilde{B}$ will be chosen large enough later and $i\geq i_{0}$.

It follows from (3.27) and (3.28) that

[TABLE]

where $C_{11}$ is a positive constant depending only on $\sup_{\Omega_{2}}(\text{tr}{h_{i}}+\text{tr}{h_{i}^{-1}})$, $C_{12}$ and $C_{13}$ are positive constants depending only on $\sup_{\Omega_{2}}(\text{tr}{h_{i}}+\text{tr}{h_{i}^{-1}})$, $\sup_{\Omega_{2}}|F_{H_{0}}|_{H_{0}}$, $\sup_{\Omega_{2}}|\theta|_{H_{0}}^{2}$, $\sup_{\Omega_{2}}|\sqrt{-1}\Lambda_{\omega}F_{H_{0}}|_{H_{0}}$, $\sup_{\Omega_{2}}|D_{H_{0}}(\Lambda_{\omega}F_{H_{0}})|_{H_{0}}^{2}$, $\sup_{\Omega_{2}}|\nabla_{H_{0}}\theta|_{H_{0}}$, $d^{-2}$ and the geometry of $(\Omega_{2},\omega)$. By choosing $\tilde{B}=e^{C_{11}}(C_{12}+1)$, then

[TABLE]

on $M$. Let $\eta_{i}(P_{0})=\sup_{M}\eta_{i}$. According to the definition of $\psi_{\alpha}$ and the uniform bounded on $\sup_{\Omega_{2}}\text{tr}{h_{i}}$ , we may assume that $P_{0}\in\Omega_{1}$. The inequality (3.31) and the maximum principle yield

[TABLE]

and then there exists a positive constant $C_{14}$ depending only on $\sup_{\Omega_{2}}(\text{tr}{h_{i}}+\text{tr}{h_{i}^{-1}})$, $\sup_{\Omega_{2}}|F_{H_{0}}|_{H_{0}}$, $\sup_{\Omega_{2}}|\theta|_{H_{0}}^{2}$, $\sup_{\Omega_{2}}|\sqrt{-1}\Lambda_{\omega}F_{H_{0}}|_{H_{0}}$, $\sup_{\Omega_{2}}|D_{H_{0}}(\Lambda_{\omega}F_{H_{0}})|_{H_{0}}^{2}$, $\sup_{\Omega_{2}}|\nabla_{H_{0}}\theta|_{H_{0}}$, $d^{-2}$ and the geometry of $(\Omega_{2},\omega)$, such that

[TABLE]

for all $i\geq i_{0}$. This concludes the proof of uniform local $C^{1}$-estimate of $H_{i}$.

We use Mochizuki’s argument ([25]) to give a uniform $L^{2}$ bounded on $|\partial_{H_{0}}h_{i}|_{H_{0}}=|\overline{\partial}_{E}h_{i}|_{H_{0}}$. Applying (2.19) and the Stokes formula, one can deduce

[TABLE]

where $\nu_{i}$ is the outer normal vector field at $\partial M_{i}$. This together with (3.27) and the uniform bounded on $\text{tr}{h_{i}}+\text{tr}{h_{i}^{-1}}$ implies that there exists a uniform positive constant $C_{15}$ such that

[TABLE]

Since we have obtained a uniform $C^{0}$-estimate and uniform local $C^{1}$-estimate on $h_{i}$, by the equation (3.26) and the standard elliptic estimates, we can derive uniform local higher order estimates on $h_{i}$. So by choosing a subsequence (which we also denote by $H_{i}$), we know that $H_{i}$ converges to a Hermitian metric $H$ on whole $M$ in $C_{loc}^{\infty}$-topology as $i\rightarrow\infty$, and $H$ satisfies the Hermitian-Einstein equation (1.2). Furthermore, (3.34) implies: $\int_{M}(|\overline{\partial}_{E}h|_{H_{0}}^{2}+|[\theta,h]|_{H_{0}}^{2})\frac{\omega^{n}}{n!}\leq C_{15}$. This complete the proof of Theorem 1.1.

4. Hermitian-Einstein metrics

In this section, we follow Mochizuki’s arguments to give a sufficient condition for the uniqueness of Hermitian-Einstein metric in Theorem 1.1.

Proposition 4.1**.**

([25], Proposition 2.4) Suppose $H_{1}$ and $H_{2}$ are two Hermitian-Einstein metrics of $(E,\overline{\partial}_{E},\theta)$. Assume (1) $H_{1}$ and $H_{2}$ are mutually bounded, (2) $\sqrt{-1}\Lambda_{\omega}F_{H_{1},\theta}=\sqrt{-1}\Lambda_{\omega}F_{H_{2},\theta}$. Then there exist a holomorphic decomposition $(E,\theta)=\bigoplus_{i=1}^{m}(E_{i},\theta|_{E_{i}})$ and a tuple $(c_{1},\ldots,c_{m})\in\mathbb{R}_{>0}^{m}$ such that

1. (i)

The decomposition $E=\bigoplus_{i=1}^{m}E_{i}$ is orthogonal with respect to both $H_{i}(i=1,2)$, 2. (ii)

$H_{1}|_{E_{i}}=c_{i}H_{2}|_{E_{i}}$.

Proof Let $h=H_{1}^{-1}H_{2}$. A direct computation gives us that

[TABLE]

From Assumption 1, it can be seen that $\sqrt{-1}\Lambda_{\omega}\partial\overline{\partial}\text{tr}{h}=0$. This means $|h^{-\frac{1}{2}}\cdot\partial_{H_{1}}h|_{H_{1}}=0$ and $|[\theta,h]\cdot h^{-\frac{1}{2}}|_{H_{1}}=0$. So $\partial_{H_{1}}h=0$ and $[\theta,h]=0$. Obviously the fact that $h$ is self-adjoint with respect to $H_{i}(i=1,2)$ implies $\overline{\partial}h=0$ and $\partial_{H_{2}}h=0$. Then it follows that the eigenvalues of $h$ are constant. Let $E=\bigoplus_{i=1}^{m}E_{i}$ denote the eigen decomposition of $h$, which is the one we desired.

Proposition 4.2**.**

Let $(M,\omega)$ be a non-compact Gauduchon manifold satisfying the assumption 1 and $|d\omega^{n-1}|_{\omega}\in L^{2}(M)$. Suppose that there is a positive exhaustion function $\phi_{1}:M\rightarrow\mathbb{R}$ such that $|d\log\phi_{1}|_{\omega}\in L^{2}(M)$. Let $(E,\overline{\partial}_{E},\theta)$ be an analytically $H_{0}$-stable bundle on $M$. Assume that $H_{i}(i=1,2)$ are Hermitian-Einstein metrics such that (1)$\det(H_{i})=\det(H_{0})$, (2) $H_{i}$ and $H_{0}$ are mutually bounded. Then $H_{1}=H_{2}$.

Proof Assume that $H_{1}$ satisfies Theorem 1.1 and let $h_{1}=H_{0}^{-1}H_{1}$.

Firstly Proposition 4.1 gives us the decomposition $(E,\theta)=\bigoplus_{j=1}^{m}(E_{j},\theta|_{E_{j}})$ such that (1) the decomposition is orthogonal with respect to $H_{i}(i=1,2)$, (2) $H_{1}|_{E_{j}}=c_{j}H_{2}|_{E_{j}}$ for some $c_{j}>0$. Let $\pi_{j}$ denote the projection onto $E_{j}$ with respect to the decomposition and $\pi_{j}^{\ast H_{0}}$ denote the adjoint of $\pi_{j}$ with respect to $H_{0}$. Because $H_{0}$ and $H_{i}(i=1,2)$ are mutually bounded, one can immediately knows that $\pi_{j}$ are bounded with respect to $H_{0}$.

Set $\partial_{H_{0},\theta}=\partial_{H_{0}}+\theta^{\ast H_{0}}$. Noting that $\theta^{\ast H_{1}}=h_{1}^{-1}\theta^{\ast H_{0}}h_{1}$ and $\partial_{H_{1}}-\partial_{H_{0}}=h_{1}^{-1}\partial_{H_{0}}h_{1}$, we deduce

[TABLE]

By Mochizuki’s arguments ([25]), we also consider the Hermitian metric $H_{3}=\bigoplus_{j=1}^{m}H_{0}|_{E_{j}}$. Then $H_{3}$ and $H_{0}$ are mutually bounded.

Set $h_{3}=H_{0}^{-1}H_{3}$. It can be expressed as that

[TABLE]

And there holds that $\overline{\partial}_{E}h_{3}\in L^{2}(H_{0})$ and $\partial_{H_{0}}h_{3}\in L^{2}(H_{0})$.

Denote $\overline{\partial}_{E,\theta}=\overline{\partial}_{E}+\theta$. Then we obtain

[TABLE]

According to the holomorphic decomposition $(E,\theta)=\bigoplus_{i=1}^{m}(E_{i},\theta|_{E_{i}})$ with respect to $H_{i}(i=1,2)$, we have $\partial_{H_{1},\theta}\pi_{j}=\partial_{H_{1}}\pi_{j}+[\theta^{\ast H_{1}},\pi_{j}]=0$ and $\overline{\partial}_{E,\theta}\pi_{j}=0$. Together with (4.2), this means

[TABLE]

Since $H_{1}$ is an Hermitian-Einstein metric on $(E,\theta)$, from Theorem 1.1, we can get that $\partial_{H_{0}}h_{1}$, $\partial_{H_{0},\theta}h_{1}$, $\partial_{H_{0}}\pi_{j}$ and $\partial_{H_{0},\theta}\pi_{j}$ are in $L^{2}(H_{0})$. Then it follows that $[\theta^{\ast H_{0}},\pi_{j}]\in L^{2}(H_{0})$ and $[\theta,\pi_{j}^{\ast H_{0}}]\in L^{2}(H_{0})$. So immediately we know $\overline{\partial}_{E,\theta}\pi_{j}^{\ast H_{0}}\in L^{2}(H_{0})$.

One can easily check that

[TABLE]

Then it can be seen that $[\theta,h_{3}]\in L^{2}(H_{0})$ and $[\theta^{\ast H_{0}},h_{3}]\in L^{2}(H_{0})$. This implies that $\overline{\partial}_{E,\theta}h_{3}$ and $\partial_{H_{0},\theta}h_{3}$ are square integrable with respect to $H_{0}$.

A direct computation gives us that

[TABLE]

and

[TABLE]

Due to the assumption $|\Lambda_{\omega}F_{H_{0},\theta}|_{H_{0}}\leq\hat{B}\varphi$, $|\Lambda_{\omega}F_{H_{0},\theta}|_{H_{0}}$ is $L^{1}$. Then we obtain the following lemma.

Lemma 4.3**.**

$\Lambda_{\omega}\text{tr}\overline{\partial}_{E,\theta}(h_{3}^{-1}\partial_{H_{0},\theta}h_{3})$* is $L^{1}$.*

Furthermore, we can derive

Lemma 4.4**.**

$\int_{M}\text{tr}(\overline{\partial}_{E,\theta}(h_{3}^{-1}\partial_{H_{0},\theta}h_{3}))\wedge\omega^{n-1}=0$.

Proof Set $\chi_{N}:=\rho(N^{-1}\phi_{1})$, where $\rho$ is a nonnegative $C^{\infty}$-function such that $\rho(t)=0$ if $t\geq 2$ and $\rho(t)=1$ if $t\leq 1$.. Because $\text{tr}(\overline{\partial}_{E,\theta}(h_{3}^{-1}\partial_{H_{0},\theta}h_{3}))$ is $L^{1}$, we just need to show

[TABLE]

Computing directly yields that

[TABLE]

Clearly $|d\omega^{n-1}|_{\omega}\in L^{2}(M)$ and $|d\log\phi_{1}|_{\omega}\in L^{2}(M)$ implies (4.9).

Combining (4.4) and Lemma 4.4, we obtain

[TABLE]

From $rankE=\sum_{j=1}^{m}rankE_{j}$, one can see that there exists $j_{0}$ such that $\mu(E,H_{0})\leq\mu(E_{j_{0}},H_{0}|_{E_{j_{0}}})$. This contradicts with the analytic stability of $(E,\overline{\partial}_{E},H_{0})$. So $H_{1}=H_{2}$.