# Poisson metrics and Higgs bundles over noncompact Kähler manifolds

Di Wu  Di Wu, School of Mathematical Sciences, University of Science and
Technology of China, Hefei 230026, People’s Republic of China
<EMAIL_ADDRESS>and  Xi Zhang  Xi Zhang, School of Mathematical
Sciences, University of Science and Technology of China, Hefei 230026,
People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the existence of Poisson metrics on flat vector
bundles over noncompact Riemannian manifolds and discuss related consequence,
specially on the applications in Higgs bundles, towards generalizing Corlette-
Donaldson-Hitchin-Simpson’s nonabelian Hodge correspondence to noncompact
Kähler manifolds setting.

###### Key words and phrases:

Flat bundle, Noncompact manifold, Analytically stability, Poisson metric,
Higgs bundle, Nonabelian Hodge correspondence

######  Mathematics Subject Classification:

53C07, 53C21

The research was partially supported by the project ”Analysis and Geometry on
Bundle” of Ministry of Science and Technology of the People’s Republic of
China, No. SQ2020YFA070080. The authors are partially supported by NSF in
China No. 11625106, 11801535 and 11721101.

##  1\. Introduction

###  1.1. Background

Let $(E,D)$ be a vector bundle over a Riemannian manifold $(M,g)$, we say
$(E,D)$ is simple if it admits no proper $D$-invariant sub-bundle and $(E,D)$
is semi-simple111In literature, one sometimes says that complete reducible or
reductive. if it is a direct sum of $(E_{i},D_{i})$ and each of them is
simple. The connection $D$ is called irreducible if it admits no nontrivial
covariantly constant section of $\operatorname{End}(E)$. Given any metric $H$
on $E$, it splits the connection as

(1.1) $\begin{split}D=D_{H}+\psi_{H},\end{split}$

where $D_{H}$ is a connection preserving $H$ and
$\psi_{H}\in\Omega^{1}(\operatorname{End}(E))$ is self-adjoint with respect to
$H$. We say $H$ is harmonic if it satisfies

(1.2) $\begin{split}D_{H}^{\ast}\psi_{H}=0,\end{split}$

where $D_{H}^{\ast}$ is the adjoint of $D_{H}$. The notion of harmonic map
originated from Eells-Sampson [17] and was generalized to the notion of
harmonic metric by Corlette [8]. The motivation for this name may be explained
as follows. By the Riemann-Hilbert correspondence, suppose
$\rho_{D}:\pi_{1}(M)\rightarrow GL(r)$ bs the representation corresponding to
a flat bundle $(E,D)$, then any metric $H$ on $E$ induces a
$\rho_{D}$-equivariant map $f_{H}$ from $\tilde{M}$ to $GL(r)/U(r)$, where
$\tilde{M}$ is the universal of $M$. Then we can see that $H$ being harmonic
is equivalent to $f_{H}$ being a harmonic map or $df_{H}$ being a harmonic
$1$-form with values in $f_{H}^{-1}T(GL(r)/U(r))$.

The study of harmonic metrics is a fundamental problem which remains an active
branch of current research in differential geometry, it builds a bridge
between geometry analysis and many other areas including group
representations, nonabelian Hodge theory. Let $(M,g)$ be a compact Riemannian
manifold, the main existence result is the following

######  Theorem 1.1 (Donaldson [14],Corlette [8]).

A flat bundle admits a harmonic metric if and only if it arises from a semi-
simple representation of the fundamental group $\pi_{1}(M)$.

If $(M,g)$ happens to be a compact Kähler manifold, the significance of this
theorem lies in the construction of Higgs bundle. In fact, the pair
$(D_{H}^{0,1},\psi_{H}^{1,0})$ will determine a poly-stable Higgs structure,
provided $H$ being harmonic. For this reason, sometimes we then say that $D$
satisfies the Hitchin’s self-duality equation [19] and we call $D$ a Hitchin
connection. Combing this with the Donaldson-Uhlenbeck-Yau theorem([13, 44])
for Higgs bundles [19, 39], one obtains Corlette-Donaldson-Hitchin-Simpson’s
nonabelian Hodge correspondence:

######  Theorem 1.2 ([8, 14, 19, 39]).

On a compact Kähler manifold, there is an one to one correspondence between
the category of seme-simple flat bundles and the category of poly-stable Higgs
bundles with vanishing Chern numbers.

This identification can be traced back to Narasimhan-Seshadri [34] and has
yielded serval fascinating topological results, specially on the fundamental
groups of compact Kähler manifolds, see [10, 41] for more details. In [21,
22], Jost-Zuo studied the representations of fundamental groups for quasi-
projective varieties by using the theory of harmonic metrics on quasi-
projective varieties. Besides that, there are also some partial existence
results for harmonic metrics on noncompact manifolds. Simpson [40] obtained
the existence of harmonic metrics on punctured Riemannian surface in his study
of nonabelian Hodge theory over noncompact curves. From the view point of
equivariant harmonic maps, Li [26] considered the case that $M=\overline{M}-D$
equipped with a Poincaré-like metric $g$, where $\overline{M}$ is a compact
complex manifold and $D$ be a divisor with normal crossings in $\overline{M}$.
Recently, Mochizuki makes a significant process towards the generations of the
Kobayashi-Hitchin correspondence and the nonabelian Hodge correspondence, the
existence of pluriharmonic metrics for good filtered flat bundles over smooth
projective varieties play an important role, see [29, 30, 31, 32]. There are
many interesting and important literature concerning the existence of harmonic
metrics and related topics, see [3, 6, 9, 12, 20, 25, 35, 48] and the
references therein.

Flat bundles occur in great abundance since it corresponds local systems and
representations of fundamental groups. It was observed by Loftin [27] that for
an affine manifold $M$, any flat bundle $(E,D)$ over $M$ corresponds a
holomorphic bundle $\tilde{E}$ over the tangent bundle $TM$. And then the
classical Donaldson-Uhlenbeck-Yau theorem can be generalized to the affine
case, see [5, 27, 38]. Recently, Collins-Jacob-Yau [7] found that the
dimension reduction of the Hermitian-Yang-Mills equation on $\tilde{E}$ is
given by the Poisson metric equation on $(E,D)$. Here we say $H$ is a Poisson
metric if

(1.3)
$\begin{split}D_{H}^{\ast}\psi_{H}=c_{H}\operatorname{id}_{E},\end{split}$

where $c_{H}$ is a function on $M$. Motivated by this and the understanding of
stable vector bundles on $K3$ surface in the large complex structure limit,
Collins-Jacob-Yau [7] focused on the research of Poisson metrics on flat
vector bundles. Under some suitable assumptions, they related the existence of
Poisson metrics on flat bundles with parabolic structures over punctured
Riemann surfaces to certain notion of slope stability.

###  1.2. Main results

In the first part of this paper, we shall attack the existence of Poisson
metrics on flat bundles over noncompact Riemannian manifolds of arbitrary
dimension. Below we state our first theorem, which can be interpreted as a
noncompact version of the result of Donaldson and Corlette.

######  Theorem 1.3.

Let $(M,g)$ be a Riemannian manifold(not necessarily compact) satisfying the
Assumptions 2.1 and either Assumption 2.2 or complete with finite volume, $E$
be a vector bundle over $M$ equipped with a connection $D$ and a metric $K$
such that $|D_{K}^{\ast}\psi_{K}|\leq C\phi$ for a constant $C$. Then it
admits a unique Poisson metric $H$ such that $\det h=1$, $|h|\in L^{\infty}$
and $|Dh|\in L^{2}$ for $h=K^{-1}H$, provided $D$ being flat and
$K$-analytically stable. Conversely, if $D$ is irreducible, it admits such a
Poisson metric only if $D$ is $K$-analytically stable.

For the existence statement, we actually only need the Assumption 2.1 and the
condition $|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$, where
$(D_{K}^{\ast}\psi_{K})^{\perp}$ is the trace-free part of
$D_{K}^{\ast}\psi_{K}$, see Proposition 3.8 for details. The Assumption 2.1 is
introduced by Mochizuki [33] for Kähler manifolds and the volume can be
infinite. The notion of analytically stability in Theorem 1.3 will be
introduced in Section 2.2. If $M$ is compact, we know that $D$ being
analytically stable(poly-stable) if and only if it is simple(semi-simple),
while the correct substitution should be replaced by analytically stability in
noncompact setting. Moreover, if $(M,g)$ be a Kähler manifold and $D$ being
flat, the definition coincides with that in Simpson’s paper [40], see Remark
2.1.

The second statement in Theorem 1.3 indicates that $D$ is analytically stable,
not only with respect to the Poisson metric, but also the background metric
$K$. It may be mentioned that it is facilitated by the feature of the
stability of vector bundles, while the issue is more complicated in the
setting of holomorphic bundles. We will use Mochizuki’s exhaustion method in
[33], where he proved the existence of exhaustion function $f$. Denote by
$M_{s}$ the set where $f\leq s$ and we firstly consider the parabolic heat
flow $(\ref{flow1})$ on $M_{s}$, which can be regard as the vector bundle
analogy of Eells-Sampson’s flow for harmonic maps or Donaldson’s heat flow
[13, 15]. For arbitrary $D$, we show the existence and uniqueness of the long-
time solution to $(\ref{flow1})$, also the Dirichlet boundary problems for
harmonic metric and Poisson metric are uniquely solvable, see Proposition 3.3
and Proposition 3.4. Next we observe in Proposition 3.5 that the crucial
identity $(\ref{outlinekeyidentity})$, based on which serval core a prior
estimates are established. Once the zeroth estimate of a sequence of the
solutions to the Dirichlet boundary problem is obtained, the evolved metrics
are completely controlled and will converge to a Poisson metric. Finally, we
prove the uniqueness and the second statement by comparing the analytically
stabilities with respect to different reference fiber metrics.

We mention that Corlette [8] studied the following heat flow

(1.4) $\begin{split}\frac{\partial D_{t}}{\partial
t}=-D_{t}D_{t,K}^{\ast}\psi_{D_{t},K},\end{split}$

on flat complex vector bundles $(E,D_{0})$ over compact Riemannian manifolds,
where $D_{t}$ is a time-dependent family of connections in the orbit of
$D_{0}$. Corlette proved the convergence of $D_{t}$ at $t=\infty$ with the
property $D_{\infty,K}\psi_{D_{\infty},K}=0$ and if $(E,D_{0})$ is simple,
$D_{\infty}$ must lie in the complex gauge orbit of $D_{0}$. Corlette’s proof
relies on Uhlenbeck’s deep compactness theorem [43] and cannot be generalized
to noncompact setting directly. Even in compact case, here the discussion is
different from that in [8] and we bypass Uhlenbeck’s compactness theorem.

As a corollary of $(\ref{DHstarpsiHt})$ and Theorem 1.3, we have

######  Corollary 1.1.

Under the same assumptions in Theorem 1.3 and
$\operatorname{tr}D_{K}^{\ast}\psi_{K}=0$, there exist a unique harmonic
metric $H$ such that $\det h=1$, $|h|\in L^{\infty}$ and $|Dh|\in L^{2}$ for
$h=K^{-1}H$, provided $D$ being flat and $K$-analytically stable. Conversely,
$D$ is $K$-analytically stable if it is irreducible and admits such a harmonic
metric.

By $(\ref{DHKstarpsiHK})$, we know that

(1.5)
$\begin{split}\operatorname{tr}D^{\ast}_{H}\psi_{H}-\operatorname{tr}D^{\ast}_{K}\psi_{K}&=-\frac{1}{2}\operatorname{tr}D_{K}^{\ast}(h^{-1}\delta_{K}h)=\frac{1}{2}\Delta\log\det
h,\end{split}$

where $h=K^{-1}H$ and $\delta_{K}=D_{K}-\psi_{K}$. So by solving a scalar
Poisson equation(Theorem 4.3 in [45]) and employing Theorem 1.3, it is easy to
conclude

######  Corollary 1.2.

Let $(M,g)$ be a Riemannian manifold satisfying the Assumption 2.1 and
Assumption 2.3 with $\phi=1$, $E$ be a vector bundle over $M$ equipped with a
connection $D$ and a metric $K$ such that $|D_{K}^{\ast}\psi_{K}|\in
L^{\infty}$, $|\psi_{K}|\in L^{2}$. Then it admits a unique harmonic metric
$H$ such that $\det h=1$, $|h|\in L^{\infty}$ and $|Dh|\in L^{2}$ for
$h=K^{-1}H$, provided $D$ being flat and $K$-analytically stable. Conversely,
$D$ is $K$-analytically stable if it is irreducible and admits such a harmonic
metric.

There is a remark on the situation that there is no stability(simple)
assumption. For an arbitrary connection on the vector bundle over a compact
Riemannian manifold, there is no obstruction to the existence of
$L^{\infty}$-approximate solutions of $(\ref{harmonicequation})$, see
Proposition 3.2.

In the second part of this paper, we shall discuss related applications and
consequences after Theorem 1.3. Recall a Higgs bundle
$(E,\overline{\partial}_{E},\theta)$ over a complex manifold $X$ consists of a
holomorphic vector bundle $(E,\overline{\partial}_{E})$ over $X$ together with
holomorphic section $\theta\in\Omega^{1,0}(\operatorname{End}(E))$ such that
$\theta\wedge\theta=0$. For a Hermitian metric $K$, we consider the Hitchin-
Simpson connection

(1.6)
$\begin{split}D_{\overline{\partial}_{E},\theta,K}=\partial_{\theta,K}+\overline{\partial}_{E,\theta},\partial_{\theta,K}=\partial_{K}+\theta^{\ast
K},\overline{\partial}_{E,\theta}=\overline{\partial}_{E}+\theta,\end{split}$

where $D_{\overline{\partial}_{E},K}=\partial_{K}+\overline{\partial}_{E}$ is
the Chern connection. If the curvature
$F_{\overline{\partial}_{E},\theta,K}=D_{\overline{\partial}_{E},\theta,K}^{2}$
vanishes, we will call $(\overline{\partial}_{E},\theta,K)$ a Higgs flat
structure on the underlying bundle $E$. In particular, if the Chern curvature
$F_{\overline{\partial}_{E},K}=D_{\overline{\partial}_{E},K}^{2}$
vanishes(that is, $\theta=0$), we then say $(\overline{\partial}_{E},K)$ a
flat holomorphic structure.

As an application of Theorem 1.3, we have

######  Theorem 1.4.

Let $(X,\omega)$ be a Kähler manifold satisfying the Assumption 2.1 and
$(E,D)$ be a $K$-analytically stable flat bundle over $X$ with
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for a constant $C$, then

  1. (1)

If $(X,\omega)$ is a complex curve, there is a Higgs flat structure on $E$.

  2. (2)

If $(X,\omega)$ is complete with bounded Ricci curvature from below and
$|\psi_{K}|\in L^{2}$, there is a Higgs flat structure on $E$.

  3. (3)

If $(X,\omega)$ is complete with nonnegative Ricci curvature and
$|\psi_{K}|\in L^{2}$, there is a flat holomorphic structure on $E$.

Especially, the third statement of the conclusions in Theorem 1.4 indicates
there is a metric compatible flat connection on $E$. Finding metric compatible
flat connections may be of interest in its own right. By Corlette’s work in
[8], it is easy to see the same conclusion also holds for a flat bundle over a
compact Riemannian manifold with nonnegative Ricci curvature and so it follows
that the Euler class $e(E)$ of $E$ is zero.

In order to prove Theorem 1.4, we need to construct Higgs structures by
employing Theorem 1.3, see Proposition 4.3 and Proposition 4.4, which is a key
step to establish the Corlette-Donaldson-Hitchin-Simpson’s correspondence. If
$X$ is compact and $H$ being harmonic, this is equivalent to determine whether
$H$ being pluriharmonic, the issue does not appear and it is essentially due
to Siu-Sampson [37, 42] and Corlette [8], see also [41]. If $X$ is noncompact,
see [22] for case that the complement of a divisor in a compact Kähler
manifold.

Secondly, we will focus on generalizing the Corlette-Donaldson-Hitchin-
Simpson’s correspondence to noncompact Kähler manifolds setting. Following
Simpson [39], if $(\overline{\partial}_{E},\theta)$ is a Higgs structure on
$E$, the $K$-analytically degree is defined by

(1.7)
$\begin{split}\deg_{\omega}(E,K)=\int_{X}\sqrt{-1}\operatorname{tr}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}\operatorname{dvol}_{\omega}.\end{split}$

If $\mathcal{V}\subseteq\mathcal{O}_{X}(E)$ is a sub-sheaf, then it can be
seen as a sub-bundle $V$ outside a singular set $\Sigma_{\mathcal{V}}$ of
codimension at least two. Let $\pi_{K}$ denotes the projection onto $V$ and
$K$ restricts to a metric on $V$ so that we can define
$\deg_{\omega}(\mathcal{V},K)$ by integrating outside $\Sigma_{\mathcal{V}}$.
We have the Chern-Weil formula written in terms of
$\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}$,

(1.8)
$\begin{split}\deg_{\omega}(\mathcal{V},K)=\int_{X\setminus\Sigma_{\mathcal{V}}}\left(\sqrt{-1}\operatorname{tr}(\pi_{K}\circ\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K})-|\overline{\partial}_{E,\theta}\pi_{K}|_{K}^{2}\right)\operatorname{dvol}_{\omega}.\end{split}$

$(\overline{\partial}_{E},\theta)$ is called $K$-analytically stable if for
any proper sub-Higgs sheaf $\mathcal{V}\subset\mathcal{O}_{X}(E)$, it holds

(1.9)
$\begin{split}\frac{\deg_{\omega}(\mathcal{V},K)}{\operatorname{rank}(\mathcal{V})}<\frac{\deg_{\omega}(E,K)}{\operatorname{rank}(E)}.\end{split}$

From now on, we fix a Hermitian metric $K$ on the vector bundle $E$, write

  * •

$\mathcal{M}_{Flat,K}$: the set of isomorphic classes of $K$-analytically
stable and irreducible flat structures such that
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\in L^{1}$.

  * •

$\mathcal{M}_{Higgs,K}$: the set of isomorphic classes of $K$-analytically
stable and irreducible Higgs structures with $c_{1}(E,K)=0$ and
$|\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}|\in L^{1}$.

  * •

$\mathcal{M}_{Flat,K,\phi}$: the set of isomorphic classes of $K$-analytically
stable and irreducible flat structures such that
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for a constant $C$.

  * •

$\mathcal{M}_{Higgs,K,\phi}$: the set of isomorphic classes of
$K$-analytically stable and irreducible Higgs structures with $c_{1}(E,K)=0$
and $|\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}|\leq C\phi$ for a
constant $C$.

In the above, isomorphic means that they are in the the same
$\mathcal{G}$-orbit, where $\mathcal{G}$ denotes the set of
$L^{\infty}$-sections of $\operatorname{Aut}(E)$. We say a Higgs structure
$(\overline{\partial}_{E},\theta)$ is irreducible if
$D_{\overline{\partial}_{E},\theta,H}$ is irreducible, for any Hermitian-
Einstein metric $H$ with $\det h=1$ and $|h|\in L^{\infty}$ for $h=K^{-1}H$.

Our third theorem can be stated as the follows.

######  Theorem 1.5.

Assume $(X,\omega)$ be a noncompact complex curve satisfying the Assumption
2.1 and either Assumption 2.2 or complete with finite volume, then we have

  1. (1)

There is map from $\mathcal{M}_{Flat,K,\phi}$ to $\mathcal{M}_{Higgs,K}$.

  2. (2)

There is map from $\mathcal{M}_{Higgs,K,\phi}$ to $\mathcal{M}_{Flat,K}$.

Notice that the complex plane $\mathbb{C}$ and compact Riemann surfaces
punctured finite points satisfy the assumptions in Theorem 1.5. Moreover, the
objects are analytically stable with respect to the same metric $K$. For
higher dimensional case, under some extra technical assumptions(see [39, 33]),
the direction that from $K$-analytically stable and irreducible Higgs
structures to $K$-analytically stable and irreducible flat structures can be
finished by the results of Simpson [39], Mochizuki [33] and the discussion as
that in $(2)$ of Theorem 1.5. Conversely, start with a $K$-analytically stable
and irreducible flat structure, using Proposition 4.4(or Proposition 4.5)
instead, we can obtain an irreducible Higgs structure. In the process, the
analytic estimates in Theorem 1.5 remain valid, but currently it is not clear
that when the resulting Higgs structure is stable with respect to $K$. We will
further discuss this problem in the future.

Theorem 1.5 is established based on the existence of Poisson metrics(Theorem
1.3) and Hermitian-Einstein metrics([33, 39]), while analytic estimates and
the comparison of different analytically stabilities play the significant
roles. If $X$ is compact, Theorem 1.5 and the corresponding result for high
dimensional case reduce to the classical nonabelian Hodge correspondence on
compact Kähler manifolds. This correspondence has been generalised to
parabolic and noncompact setting, specially on the case of a projective
variety equipped with normal crossing divisors, in a series of works including
those of Simpson [40] and Biquard-Boalch [2] for the tame and wild case in
complex dimension one. And the higher dimensional case can be found in [1, 29,
31, 32].

###  1.3. Organization

The rest of this paper is organized as follows. In Section 2, we discuss the
assumptions on the base manifolds, introduce the notation of stability of
vector bundles, and present some useful calculations. In Section 3, we solve
the Dirichlet boundary problems for harmonic metric equation and Poisson
metric equation, then give the proof of Theorem 1.3. In Section 4, we firstly
investigate the uniqueness of Poisson metrics and then prove Theorem 1.4,
Theorem 1.5. As a byproduct, also a vanishing theorem of Kamber-Tondeur
classes is obtained.

##  2\. Preliminaries

###  2.1. Assumptions on the base spaces

Assume $(M,g)$ be a Riemannian manifold, we list the following assumptions
that needed in this paper.

######  Assumption 2.1.

There is a function $\phi:M\rightarrow\mathbb{R}_{\geq 0}$ with $\phi\in
L^{1}$, such that if $f$ is a nonnegative bounded function on $M$ satisfying
$\Delta f\geq-B\phi$ for a positive constant $B$ in distribution sense, we
have

(2.1) $\begin{split}\sup\limits_{M}f\leq
C(B)(1+\int_{M}f\phi\operatorname{dvol}_{g}).\end{split}$

Furthermore, if $f$ satisfies $\Delta f\geq 0$, we have $\Delta f=0$.

######  Assumption 2.2.

$(M,g)$ admits an exhaustion function $\rho:M\rightarrow\mathbb{R}_{\geq 0}$
with $|\Delta\rho|\in L^{1}$.

######  Assumption 2.3.

$(M,g)$ admits an exhaustion function $\rho:M\rightarrow\mathbb{R}_{\geq 0}$
with bounded $\Delta\rho$.

The above assumptions are introduced for Kähler manifolds in [33] and [39].

######  Example 2.1.

Compact Riemannian manifolds satisfy Assumptions 2.1, 2.2 and 2.3.

######  Example 2.2.

Let $M$ be a Zariski open subset of a compact Kähler manifold
$(\overline{X},\overline{g})$, $g$ is the restriction of $\overline{g}$, then
it satisfies Assumptions 2.1, 2.2 and 2.3.

######  Example 2.3.

Consider $(\mathbb{R}^{2},g_{\mathbb{R}^{2}})$ with the Euclidean metric
$g_{\mathbb{R}^{2}}$ and
$\phi_{\mathbb{R}^{2}}=(1+|r|^{2})^{-1-\delta},\delta>0$, then it has infinite
volume and satisfies Assumptions 2.1. Moreover,
$(\mathbb{R}^{2},\phi_{\mathbb{R}^{2}}g_{\mathbb{R}^{2}})$ satisfies
Assumptions 2.1 and has finite volume.

######  Example 2.4.

Any product manifold $(\mathbb{R}^{2},g_{\mathbb{R}^{2}})\times(N,g_{N})$ has
infinite volume and satisfies Assumptions 2.1, where $(N,g_{N})$ is a compact
Riemannian manifold.

###  2.2. Stability of vector bundles

Assume $(E,D)$ be a vector bundle over a Riemannian manifold $(M,g)$, equipped
with a metric $K$ on $E$.

######  Definition 2.1.

The $K$-analytically degree of $(E,D)$ is defined by

(2.2)
$\begin{split}\deg_{g}(E,D,K)=-\int_{M}\operatorname{tr}D_{K}^{\ast}\psi_{K}\operatorname{dvol}_{g}.\end{split}$

For any $D$-invariant sub-bundle $S\subseteq E$, we define

(2.3)
$\begin{split}\deg_{g}(S,D,K)=-\int_{M}\operatorname{tr}D_{K_{S}}^{\ast}\psi_{K_{S}}\operatorname{dvol}_{g},\end{split}$

where $K_{S}$ is the restricted metric on $S$. Then $(E,D)$ is said to be
$K$-analytically stable if for any nontrivial $D$-invariant sub-bundle $S$, it
holds that

(2.4)
$\begin{split}\frac{\deg_{g}(S,D,K)}{\operatorname{rank}(S)}<\frac{\deg_{g}(E,D,K)}{\operatorname{rank}(E)}.\end{split}$

######  Remark 2.1.

If $(M,\omega)$ be a Kähler manifold and $D$ is a flat connection, we have

(2.5)
$\begin{split}\deg_{g}(E,D,K)=\deg_{\omega}(E,D^{0,1},K)=-\deg_{\omega}(E,\delta_{K}^{0,1},K),\end{split}$

where $\delta_{K}^{0,1}=D_{K}^{0,1}-\psi_{K}^{0,1}$,
$\deg_{\omega}(E,D^{0,1},K)$, $\deg_{\omega}(E,\delta_{K}^{0,1},K)$ are the
$K$-analytically degree of the holomorphic bundles $(E,D^{0,1})$ and
$(E,\delta_{K}^{0,1})$ respectively.

For any $D$-invariant sub-bundle $S$, we have the Chern-Weil formula written
in terms of $D_{K}^{\ast}\psi_{K}$,

(2.6)
$\begin{split}\deg_{g}(S,D,K)=-\int_{M}\left(\operatorname{tr}(\pi_{K}\circ
D_{K}^{\ast}\psi_{K})+\frac{1}{2}|D\pi_{K}|^{2}_{K}\right)\operatorname{dvol}_{g},\end{split}$

where $\pi_{K}$ is the orthogonal projection onto $S$.

######  Proposition 2.1.

If a vector bundle $(E,D)$ admits a Poisson metric $K$, it is an orthogonal
direct sum of $K$-analytically stable bundles.

###### Proof.

For any nontrivial $D$-invariant sub-bundle $S\subset E$, the Chern-Weil
formula yields

(2.7)
$\begin{split}\frac{\deg_{g}(S,D,K)}{\operatorname{rank}(S)}&=\frac{-\int_{M}\left(\operatorname{tr}(\pi_{K}\circ
D_{K}^{\ast}\psi_{K})+\frac{1}{2}|D\pi_{K}|^{2}_{K}\right)\operatorname{dvol}_{g}}{\operatorname{rank}(S)}\\\
&=\frac{-\int_{M}\left(\frac{\operatorname{rank}(S)}{\operatorname{rank}(E)}\operatorname{tr}D_{K}^{\ast}\psi_{K}+\frac{1}{2}|D\pi_{K}|^{2}_{K}\right)\operatorname{dvol}_{g}}{\operatorname{rank}(S)}\\\
&\leq\frac{\deg_{g}(E,D,K)}{\operatorname{rank}(E)}.\end{split}$

If the equality holds, we obtain $D\pi_{K}=0$. Let $S^{\perp}$ denote the
orthogonal complement of $S$ in $E$ with respect to $K$, then it is also a
$D$-invariant sub-bundle such that
$(E,D)\cong(S,D_{S})\oplus(S^{\perp},D_{S^{\perp}})$, where $D_{S}$ and
$D_{S^{\perp}}$ are the induced connections. Moreover, the induced metrics
$K_{S}$ and $K_{S^{\perp}}$ both are Poisson metrics and we complete the proof
by an easy induction. ∎

###  2.3. Some useful calculations

We present some important calculations, it is emphasized that we are not
working under the local flat frame and hence most of them need not the
assumption that the underlying connection is flat.

Given a vector bundle $E$ over a Riemannian manifold $(M,g)$, the exterior
differential operator $D:\Omega^{p}(E)\rightarrow\Omega^{p+1}(E)$ relative to
a connection is given by

(2.8)
$\begin{split}D\omega(e_{0},...,e_{p})&=\sum\limits_{k=0}^{p}(-1)^{k}D_{e_{k}}(\omega(e_{0},e_{1},...,\hat{e_{k}},...,e_{p}))\\\
&+\sum\limits_{k<l}(-1)^{k+l}\omega([e_{k},e_{l}],e_{1},...,\hat{e_{k}},...,\hat{e_{l}},...,e_{p}).\end{split}$

Since the Levi-Civita connection on $TM$ is torsion-free, we also have

(2.9)
$\begin{split}D\omega(e_{0},...,e_{p})&=\sum\limits_{k=0}^{p}(-1)^{k}(\tilde{\nabla}_{e_{k}}\omega)(e_{0},...,\hat{e_{k}},...,e_{p}),\end{split}$

where $\tilde{\nabla}$ is the induced connection on $\Omega^{\ast}(E)$.

For a metric $K$ on $E$, we define the point-wise inner product

(2.10)
$\begin{split}K(\omega,\theta)(x)=\sum\limits_{i_{1}<...<i_{p}}(\omega(e_{i_{1}},...,e_{i_{p}}),\theta(e_{i_{1}},...,e_{i_{p}}))_{K},\end{split}$

where $\omega,\theta\in\Omega^{p}(E)$ and $\\{e_{i}\\}_{i=1}^{\dim M}$ is the
orthogonal unit basis of $T_{x}M$. Then the decomposition
$(\ref{decomposition})$ is achieved by choosing $\psi_{K}$ such that

(2.11)
$\begin{split}K(\psi_{K}(X),Y)=\frac{1}{2}(K(DX,Y)+K(X,DY)-dK(X,Y)),\end{split}$

for any $X,Y\in\Omega^{0}(E)$. With respect to the Riemannian structures of
$E$ and $TM$, the co-differential operator
$D_{K}^{\ast}:\Omega^{p}(E)\rightarrow\Omega^{p-1}(E)$ is characterized via
the formula

(2.12)
$\begin{split}\int_{M}K(D_{K}\omega,\theta)\operatorname{dvol}_{g}=\int_{M}K(\omega,D_{K}^{\ast}\theta)\operatorname{dvol}_{g},\end{split}$

for any $\omega\in\Omega^{p-1}(E),\theta\in\Omega^{p}(E)$ and either $\omega$
or $\theta$ is compactly supported. Then it holds

(2.13)
$\begin{split}D_{K}^{\ast}\theta(e_{1},..,e_{p-1})=-\sum\limits_{i}(\nabla_{K,e_{i}}\theta)(e_{i},e_{1},...,e_{p-1}),\end{split}$

where $\nabla_{K}$ is the connection on $\Omega^{\ast}(E)$, induced by the
Levi-Civita connection $\nabla$ and $D_{K}$.

If $H$ be another metric on $E$, we define the positive endomorphism
$h=K^{-1}H$ by setting $H(\cdot,\cdot)=K(h(\cdot),\cdot)$. A straightforward
calculation, using $(\ref{psi})$, shows that

(2.14)
$\begin{split}D_{H}&=D_{K}+\frac{1}{2}h^{-1}\delta_{K}h=\frac{1}{2}(D_{K}+h^{-1}\circ
D_{K}\circ h+\psi_{K}-h^{-1}\circ\psi_{K}\circ h),\end{split}$ (2.15)
$\begin{split}\psi_{H}&=\psi_{K}-\frac{1}{2}h^{-1}\delta_{K}h=\frac{1}{2}(\psi_{K}+h^{-1}\circ\psi_{K}\circ
h+D_{K}-h^{-1}\circ D_{K}\circ h),\end{split}$ (2.16)
$\begin{split}\delta_{H}&=\delta_{K}+h^{-1}\delta_{K}h=h^{-1}\circ\delta_{K}\circ
h,\end{split}$

where $\delta_{K}=D_{K}-\psi_{K}$.

######  Lemma 2.1.

If $H(t)$ is a one-parameter family of fiber metrics on the vector bundle
$(E,D)$, then we have

(2.17) $\begin{split}\frac{\partial D_{H(t)}}{\partial
t}=\frac{1}{2}D_{H(t)}(h^{-1}(t)\frac{\partial h(t)}{\partial
t})-\frac{1}{2}[\psi_{H}(t),h^{-1}(t)\frac{\partial h(t)}{\partial
t}],\end{split}$ (2.18) $\begin{split}\frac{\partial\psi_{H(t)}}{\partial
t}=-\frac{1}{2}D_{H(t)}(h^{-1}(t)\frac{\partial h(t)}{\partial
t})+\frac{1}{2}[\psi_{H}(t),h^{-1}(t)\frac{\partial h(t)}{\partial
t}],\end{split}$ (2.19) $\begin{split}\frac{\partial
D_{H(t)}^{\ast}\psi_{H(t)}}{\partial
t}&=-\frac{1}{2}D_{H(t)}^{\ast}D_{H(t)}(h^{-1}(t)\frac{\partial h(t)}{\partial
t})+\frac{1}{2}[D_{H(t)}^{\ast}\psi_{H(t)},h^{-1}(t)\frac{\partial
h(t)}{\partial t}]\\\ &-\frac{1}{2}g^{ij}[\psi_{H(t)}(\frac{\partial}{\partial
x^{i}}),[\psi_{H(t)}(\frac{\partial}{\partial x^{j}}),h^{-1}(t)\frac{\partial
h(t)}{\partial t}]],\end{split}$

where $h(t)=K^{-1}H(t)$.

###### Proof.

It suffice to prove the $(\ref{DHt})$ and $(\ref{DHstarpsiHt})$. By making use
of $(\ref{DHDK})$, $(\ref{psiHpsiK})$, we deduce

(2.20) $\begin{split}\frac{\partial D_{H}}{\partial
t}&=\frac{1}{2}\frac{\partial}{\partial
t}(h^{-1}D_{K}h-h^{-1}\circ\psi_{K}\circ h)\\\
&=-\frac{1}{2}h^{-1}\frac{\partial h}{\partial
t}h^{-1}D_{K}h+\frac{1}{2}h^{-1}D_{K}\frac{\partial h}{\partial
t}+\frac{1}{2}h^{-1}\frac{\partial h}{\partial
t}h^{-1}\psi_{K}h-\frac{1}{2}h^{-1}\psi_{K}\frac{\partial h}{\partial t}\\\
&=-\frac{1}{2}h^{-1}\frac{\partial h}{\partial
t}h^{-1}D_{K}h+\frac{1}{2}h^{-1}D_{K}\frac{\partial h}{\partial
t}-\frac{1}{2}[\psi_{H},h^{-1}\frac{\partial h}{\partial t}]\\\
&+\frac{1}{2}h^{-1}\frac{\partial h}{\partial
t}(\psi_{H}-\psi_{K}-D_{K}+h^{-1}\circ D_{K}\circ h)\\\
&-\frac{1}{2}(\psi_{H}-\psi_{K}-D_{K}+h^{-1}\circ D_{K}\circ
h)h^{-1}\frac{\partial h}{\partial t}\\\
&=\frac{1}{2}D_{H}(h^{-1}\frac{\partial h}{\partial
t})-\frac{1}{2}[\psi_{H},h^{-1}\frac{\partial h}{\partial
t}]-\frac{1}{2}h^{-1}\frac{\partial h}{\partial t}h^{-1}D_{K}h\\\
&+\frac{1}{2}h^{-1}D_{K}\frac{\partial h}{\partial
t}+\frac{1}{2}[h^{-1}\frac{\partial h}{\partial t},h^{-1}\circ D_{K}\circ
h]\\\ &=\frac{1}{2}D_{H}(h^{-1}\frac{\partial h}{\partial
t})-\frac{1}{2}[\psi_{H},h^{-1}\frac{\partial h}{\partial t}].\end{split}$

On the other hand, let’s choose the local normal coordinate
$(x^{1},...,x^{n})$ at the considered point, using $(\ref{DHt})$ and
$(\ref{psiHt})$, we have

(2.21) $\begin{split}\frac{\partial D_{H}^{\ast}\psi_{H}}{\partial
t}&=-\frac{\partial}{\partial t}[D_{H,\frac{\partial}{\partial
x^{i}}},\psi_{H}(\frac{\partial}{\partial x^{i}})]\\\
&=-[\frac{\partial}{\partial t}D_{H,\frac{\partial}{\partial
x^{i}}},\psi_{H}(\frac{\partial}{\partial
x^{i}})]-[D_{H,\frac{\partial}{\partial x^{i}}},\frac{\partial}{\partial
t}\psi_{H}(\frac{\partial}{\partial x^{i}})]\\\
&=-\frac{1}{2}[D_{H,\frac{\partial}{\partial x^{i}}}(h^{-1}\frac{\partial
h}{\partial t})-[\psi_{H}(\frac{\partial}{\partial
x^{i}}),h^{-1}\frac{\partial h}{\partial t}],\psi_{H}(\frac{\partial}{\partial
x^{i}})]\\\ &+\frac{1}{2}[D_{H,\frac{\partial}{\partial
x^{i}}},D_{H,\frac{\partial}{\partial x^{i}}}(h^{-1}\frac{\partial h}{\partial
t})-[\psi_{H}(\frac{\partial}{\partial x^{i}}),h^{-1}\frac{\partial
h}{\partial t}]]\\\ &=\frac{1}{2}D_{H,\frac{\partial}{\partial
x^{i}}}D_{H,\frac{\partial}{\partial x^{i}}}(h^{-1}\frac{\partial h}{\partial
t})-\frac{1}{2}[\psi_{H}(\frac{\partial}{\partial
x^{i}}),[\psi_{H}(\frac{\partial}{\partial x^{i}}),h^{-1}\frac{\partial
h}{\partial t}]]\\\ &-\frac{1}{2}[D_{H,\frac{\partial}{\partial
x^{i}}}(h^{-1}\frac{\partial h}{\partial t}),\psi_{H}(\frac{\partial}{\partial
x^{i}})]-\frac{1}{2}[D_{H,\frac{\partial}{\partial
x^{i}}},[\psi_{H}(\frac{\partial}{\partial x^{i}}),h^{-1}\frac{\partial
h}{\partial t}]]\\\ &=-\frac{1}{2}D_{H}^{\ast}D_{H}(h^{-1}\frac{\partial
h}{\partial t})+\frac{1}{2}[D_{H}^{\ast}\psi_{H},h^{-1}\frac{\partial
h}{\partial t}]\\\ &-\frac{1}{2}[\psi_{H}(\frac{\partial}{\partial
x^{i}}),[\psi_{H}(\frac{\partial}{\partial x^{i}}),h^{-1}\frac{\partial
h}{\partial t}]].\end{split}$

∎

Under the local coordinate $(x^{1},...,x^{n})$, by using $(\ref{DHDK})$ and
$(\ref{psiHpsiK})$, it follows

(2.22)
$\begin{split}D_{H}^{\ast}\psi_{H}&=D_{K}^{\ast}\psi_{K}-\frac{1}{2}D_{K}^{\ast}(h^{-1}\delta_{K}h)-\frac{1}{2}g^{ij}[h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,\psi_{K}(\frac{\partial}{\partial x^{j}})]\\\
&=D_{K}^{\ast}\psi_{K}+\frac{1}{2}g^{ij}D_{\frac{\partial}{\partial
x^{i}}}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{j}}}h)-\frac{1}{2}g^{ij}\Gamma_{ij}^{k}h^{-1}\delta_{K,\frac{\partial}{\partial
x^{k}}}h.\end{split}$

From now on, the norms below are taken with respect to $K$ unless indicated
explicitly. Based on $(\ref{DHKstarpsiHK})$, we prove the following lemma.

######  Lemma 2.2.

Let $H$ and $K$ be two fiber metrics on the vector bundle $(E,D)$, it holds

(2.23)
$\begin{split}(D_{H}^{\ast}\psi_{H}-D_{K}^{\ast}\psi_{K},h)=\frac{1}{2}\Delta\operatorname{tr}h-\frac{1}{2}|h^{-\frac{1}{2}}\delta_{K}h|^{2},\end{split}$
(2.24)
$\begin{split}(D_{K}^{\ast}\psi_{K}-D_{H}^{\ast}\psi_{H},h^{-1})_{H}=\frac{1}{2}\Delta\operatorname{tr}h^{-1}-\frac{1}{2}|h^{\frac{1}{2}}\delta_{H}h^{-1}|_{H}^{2},\end{split}$

where $h=K^{-1}H$.

###### Proof.

We only proof $(\ref{keylemma1})$ and $(\ref{keylemma2})$ follows from
$(\ref{keylemma1})$. Choose the local normal coordinate $(x^{1},...,x^{n})$ at
the considered point, it follows

(2.25)
$\begin{split}(D_{H}^{\ast}\psi_{H}-D_{K}^{\ast}\psi_{K},h)&=-\frac{1}{2}(D_{K}^{\ast}(h^{-1}\delta_{K}h)+[h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,\psi_{K}(\frac{\partial}{\partial x^{i}})],h)\\\
&=\frac{1}{2}(D_{K,\frac{\partial}{\partial
x^{i}}}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h),h)-\frac{1}{2}([h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,\psi_{K}(\frac{\partial}{\partial x^{i}})],h)\\\
&=\frac{1}{2}\frac{\partial}{\partial
x^{i}}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,h)-\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,D_{K,\frac{\partial}{\partial x^{i}}}h)\\\
&+\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,[\psi_{K}(\frac{\partial}{\partial x^{i}}),h])\\\
&=\frac{1}{2}\frac{\partial}{\partial
x^{i}}\operatorname{tr}\delta_{K,\frac{\partial}{\partial
x^{i}}}h-\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,\delta_{K,\frac{\partial}{\partial x^{i}}}h)\\\
&=\frac{1}{2}\Delta\operatorname{tr}h-\frac{1}{2}|h^{-\frac{1}{2}}\delta_{K}h|^{2}.\end{split}$

∎

Following [13], the Donaldson’s distance on the space of metrics is defined by

(2.26)
$\begin{split}\sigma(K,H)=\operatorname{tr}(K^{-1}H)+\operatorname{tr}(H^{-1}K)-2\operatorname{rank}(E).\end{split}$

It is obvious that $\sigma(H,K)\geq 0$ with equality if and only if $H=K$.

By Lemma 2.2, we get

######  Corollary 2.1.

Let $H$ and $K$ be two harmonic metrics on the vector bundle $(E,D)$, then we
have

(2.27)
$\begin{split}\Delta\sigma(H,K)=|h^{-\frac{1}{2}}\delta_{K}h|^{2}+|h^{\frac{1}{2}}\delta_{H}h^{-1}|_{H}^{2},\end{split}$

where $h=K^{-1}H$. In particular, if $(E,D)$ is simple and $(M,g)$ satisfies
the Assumption 2.1, the mutually bounded harmonic metrics are unique up to
scaling.

Now we consider the evolution equation

(2.28) $\begin{split}H^{-1}(t)\frac{\partial H(t)}{\partial
t}=2D^{\ast}_{H(t)}\psi_{H(t)},\end{split}$

a direct calculation yields

######  Lemma 2.3.

Let $(E,D)$ be a vector bundle over $(M,g)$, assume $H(t)$ and $K(t)$ be two
solutions of the heat flow $(\ref{outlineflow})$, then we have

(2.29) $\begin{split}(\frac{\partial}{\partial
t}-\Delta)\sigma(H(t),K(t))=-|\tilde{h}(t)^{-\frac{1}{2}}\delta_{K(t)}\tilde{h}(t)|^{2}_{K(t)}-|\tilde{h}^{\frac{1}{2}}(t)\delta_{H(t)}\tilde{h}^{-1}(t)|^{2}_{H(t)},\end{split}$

where $\tilde{h}(t)=K^{-1}(t)H(t)$.

######  Lemma 2.4.

Let $(E,D)$ be a vector bundle over $(M,g)$, along the heat flow
$(\ref{outlineflow})$ it holds

(2.30)
$\begin{split}\frac{d}{dt}||\psi_{H}||_{H,L^{2}}^{2}&=-2||D_{H}^{\ast}\psi_{H}||_{H,2}^{2}-2\int_{M}d\eta_{H},\end{split}$
(2.31)
$\begin{split}\frac{d}{dt}||D_{H}^{\ast}\psi_{H}||_{H,L^{2}}^{2}=-2||D_{H}D_{H}^{\ast}\psi_{H}||_{H,L^{2}}^{2}-2||[\psi_{H},D_{H}^{\ast}\psi_{H}]||_{H,L^{2}}^{2}+2\int_{M}d\tilde{\eta}_{H},\end{split}$
(2.32) $\begin{split}(\frac{\partial}{\partial
t}-\Delta)|D_{H}^{\ast}\psi_{H}|_{H}^{2}&=-2|D_{H}D_{H}^{\ast}\psi_{H}|_{H}^{2}-2|[\psi_{H},D_{H}^{\ast}\psi_{H}]|_{H}^{2},\end{split}$

where $\eta_{H}=H(D_{H}^{\ast}\psi_{H},\ast\psi_{H})$,
$\tilde{\eta}_{H}=H(D_{H}^{\ast}\psi_{H},\ast D_{H}D_{H}^{\ast}\psi_{H})$.
Furthermore, if $D$ is a flat connecton, we have

(2.33) $\begin{split}(\frac{\partial}{\partial
t}-\Delta)|\psi_{H}|_{H}^{2}=-|[\psi_{H},\psi_{H}]|^{2}_{H}-2(\psi_{H}\circ\operatorname{Ric},\psi_{H})_{H}-2|\nabla_{H}\psi_{H}|^{2}_{H},\end{split}$

where $\operatorname{Ric}$ is the Ricci transformation of $(M,g)$.

###### Proof.

Firstly, applying the equality $(\ref{psiHt})$ yields

(2.34)
$\begin{split}\frac{d}{dt}||\psi_{H}||_{H,L^{2}}^{2}&=2\int_{M}(-\frac{1}{2}D_{H}(H^{-1}\frac{\partial
H}{\partial t})+\frac{1}{2}[\psi_{H},H^{-1}\frac{\partial H}{\partial
t}],\psi_{H})_{H}\operatorname{dvol}_{g}\\\
&=-2\int_{M}(D_{H}D_{H}^{\ast}\psi_{H},\psi_{H})_{H}\operatorname{dvol}_{g}\\\
&=-2||D_{H}^{\ast}\psi_{H}||_{H,2}^{2}-2\int_{M}d\eta_{H}.\end{split}$

Next, $(\ref{DHstarpsiHt})$ indicates that

(2.35) $\begin{split}\frac{\partial D_{H}^{\ast}\psi_{H}}{\partial
t}=-D_{H}^{\ast}D_{H}D_{H}^{\ast}\psi_{H}-g^{ij}[\psi_{H}(\frac{\partial}{\partial
x^{i}}),[\psi_{H}(\frac{\partial}{\partial
x^{j}}),D_{H}^{\ast}\psi_{H}]],\end{split}$

as it is easy to check $D_{H}^{\ast}\psi_{H}$ is self-adjoint with respect to
$H$, it then follows

(2.36)
$\begin{split}\frac{d}{dt}||D_{H}^{\ast}\psi_{H}||_{H,L^{2}}^{2}&=-2\int_{M}(D_{H}^{\ast}D_{H}D_{H}^{\ast}\psi_{H},D_{H}^{\ast}\psi_{H})_{H}\operatorname{dvol}_{g}\\\
&-2\int_{M}g^{ij}([\psi_{H}(\frac{\partial}{\partial
x^{i}}),[\psi_{H}(\frac{\partial}{\partial
x^{j}}),D_{H}^{\ast}\psi_{H}]],D_{H}^{\ast}\psi_{H})_{H}\operatorname{dvol}_{g}\\\
&=-2||D_{H}D_{H}^{\ast}\psi_{H}||_{H,L^{2}}^{2}-2||[\psi_{H},D_{H}^{\ast}\psi_{H}]||_{H,L^{2}}^{2}+2\int_{M}d\tilde{\eta}_{H},\end{split}$

and

(2.37) $\begin{split}(\frac{\partial}{\partial
t}-\Delta)|D_{H}^{\ast}\psi_{H}|_{H}^{2}&=-2(D_{H}^{\ast}D_{H}D_{H}^{\ast}\psi_{H},D_{H}^{\ast}\psi_{H})_{H}\\\
&-2g^{ij}([\psi_{H}(\frac{\partial}{\partial
x^{i}}),[\psi_{H}(\frac{\partial}{\partial
x^{j}}),D_{H}^{\ast}\psi_{H}]],D_{H}^{\ast}\psi_{H})_{H}\\\
&+2(D_{H}^{\ast}D_{H}D_{H}^{\ast}\psi_{H},D_{H}^{\ast}\psi_{H})_{H}-2|D_{H}D_{H}^{\ast}\psi_{H}|_{H}^{2}\\\
&=-2|D_{H}D_{H}^{\ast}\psi_{H}|_{H}^{2}-2|[\psi_{H},D_{H}^{\ast}\psi_{H}]|_{H}^{2}.\end{split}$

On the other hand, we have

(2.38) $\begin{split}(\frac{\partial}{\partial
t}-\Delta)|\psi_{H}|_{H}^{2}&=2(-\frac{1}{2}D_{H}(H^{-1}\frac{\partial
H}{\partial t})+\frac{1}{2}[\psi_{H},H^{-1}\frac{\partial H}{\partial
t}],\psi_{H})_{H}\\\
&+2(\nabla_{H}^{\ast}\nabla_{H}\psi_{H},\psi_{H})_{H}-2|\nabla_{H}\psi_{H}|_{H}^{2}\\\
&=-2(D_{H}D_{H}^{\ast}\psi_{H}-\nabla_{H}^{\ast}\nabla_{H}\psi_{H},\psi_{H})_{H}-2|\nabla_{H}\psi_{H}|_{H}^{2},\end{split}$

Now using the flatness of $D$: $D_{H}\psi_{H}=0$ and
$D_{H}^{2}=-\frac{1}{2}[\psi_{H},\psi_{H}]$, it is equal to

(2.39) $\begin{split}&2g^{ij}(D_{H}^{2}(\cdot,\frac{\partial}{\partial
x^{i}})(\psi_{H}(\frac{\partial}{\partial
x^{j}}))-\psi_{H}(R_{\nabla}(\cdot,\frac{\partial}{\partial
x^{i}})(\frac{\partial}{\partial
x^{j}})),\psi_{H})_{H}-2|\nabla_{H}\psi_{H}|_{H}^{2}\\\
&=-(g^{ij}[[\psi_{H},\psi_{H}(\frac{\partial}{\partial
x^{i}})],\psi_{H}(\frac{\partial}{\partial
x^{j}})]+2\psi_{H}\circ\operatorname{Ric},\psi_{H})_{H}-2|\nabla_{H}\psi_{H}|_{H}^{2}\\\
&=-|[\psi_{H},\psi_{H}]|^{2}_{H}-2(\psi_{H}\circ\operatorname{Ric},\psi_{H})_{H}-2|\nabla_{H}\psi_{H}|^{2}_{H}.\end{split}$

∎

##  3\. Analytically stability and existence of Poisson metrics

Given a vector bundle $(E,D)$ over a compact Riemannian manifold $(X,g)$,
consider the following heat flow

(3.1) $\begin{split}\left\\{\begin{array}[]{ll}H^{-1}\frac{\partial
H}{\partial t}=2D^{\ast}_{H}\psi_{H},\\\ H(0)=K.\end{array}\right.\end{split}$

And if $X$ has nonempty smooth boundary $\partial X$, for any given compatible
data $\tilde{H}$, we impose the boundary condition $H(t)|_{\partial
X}=\tilde{H}$ in the above system.

With the calculations in the last sections in hand and following [13], we have

######  Proposition 3.1.

Let $(E,D)$ be a vector bundle over a compact Riemannian manifold $(X,g)$(with
possibly nonempty boundary), then heat flow $(\ref{flow1})$ admits unique
solution defined on $[0,\infty)$.

######  Proposition 3.2.

Assume $(E,D)$ be a vector bundle over a compact Riemannian manifold $(M,g)$,
then for any $\epsilon>0$, there exists a fiber metric $H_{\epsilon}$ such
that $||D_{H_{\epsilon}}\psi_{H_{\epsilon}}||_{L^{\infty}}\leq\epsilon$.

###### Proof.

Let $H=H(t)$ be the solution to $(\ref{flow1})$, defined on $[0,+\infty)$.
Since $(\ref{flow1})$ is the negative gradient flow of the energy
$||\psi_{H}||_{H,L^{2}}$, by $(\ref{timemodule})$ we know
$||D_{H}^{\ast}\psi_{H}||_{H,L^{2}}^{2}\rightarrow 0$, as $t$ goes infinity.
Then the proof can be finished by using $(\ref{modulet})$ and parabolic
Moser’s iteration. ∎

######  Proposition 3.3.

Let $(E,D)$ be a vector bundle over a compact Riemannian manifold $(X,g)$ with
smooth boundary $\partial X$. Then for any given data $K$, there exists a
unique harmonic metric $H$ on $(E,D)$ such that $H=K$ on $\partial X$.

###### Proof.

Due to Corollary 2.1, it remain to prove the existence statement. By the
standard elliptic equation theory, let $u$ be the solution to the following
Dirichlet problem:

(3.2) $\begin{split}\Delta u=-|D_{K}^{\ast}\psi_{K}|,u|_{\partial
X}=0.\end{split}$

Proposition 3.1 ensures the existence of long-time solution $H(t)$ of
$(\ref{flow1})$ with boundary condition $H(t)|_{\partial X}=K$. Set

(3.3)
$\begin{split}v(x,t)=\int_{0}^{t}|D_{H(s)}^{\ast}\psi_{H(s)}|_{H(s)}ds-u,\end{split}$

by $(\ref{modulet})$ it is easy to see

(3.4) $\begin{split}\left\\{\begin{array}[]{ll}(\frac{\partial}{\partial
t}-\Delta)v(x,t)\leq 0,\\\ v(x,0)=-u(x),\\\ v(x,t)|_{\partial
X}=0.\end{array}\right.\end{split}$

Then the maximum principle gives

(3.5)
$\begin{split}\int_{0}^{t}|D_{H(s)}^{\ast}\psi_{H(s)}|_{H(s)}ds\leq\max\limits_{X}u.\end{split}$

Now for any $t_{1}\leq t\leq t_{2}$ and
$\tilde{h}(t)=H^{-1}(t_{1})H(t),\hat{h}(t)=H^{-1}(t)H(t_{1})$, it is
straightforward to check

(3.6) $\begin{split}&|\frac{\partial}{\partial
t}\log\operatorname{tr}\tilde{h}(t)|\leq
2|D_{H(t)}^{\ast}\psi_{H(t)}|_{H(t)},|\frac{\partial}{\partial
t}\log\operatorname{tr}\hat{h}(t)|\leq
2|D_{H(t)}^{\ast}\psi_{H(t)}|_{H(t)}.\end{split}$

A simple integration yields

(3.7) $\begin{split}\sigma(H(t_{1}),H(t_{2}))\leq
2\operatorname{rank}(E)(e^{2\int_{t_{1}}^{t_{2}}|D_{H(s)}^{\ast}\psi_{H(s)}|_{H(s)}ds}-1).\end{split}$

So by combining $(\ref{dirchlet1})$ and $(\ref{dirchlet2})$, we know that
$H(t)$ converge in $C^{0}$-topology to $H_{\infty}$ as $t\rightarrow\infty$.
Using $(\ref{dirchlet1})$ and arguing as Lemma 3.3 in [47], the elliptic
regularity theory implies that there exists a subsequence $H(t_{i})$
converging to $H_{\infty}$ in $C^{\infty}$-topology and $H_{\infty}$ is a
harmonic metric satisfying the desired boundary condition. ∎

######  Proposition 3.4.

Let $(E,D)$ be a vector bundle over a compact Riemannian manifold $(X,g)$ with
smooth boundary $\partial X$. Then for any given data $K$, there exists a
unique Poisson metric $H$ on $(E,D)$ such that $H=K$ on $\partial X$ and $\det
H=\det K$.

###### Proof.

Due to Proposition 3.3, there exists a unique harmonic metric $\tilde{H}$ such
that $\tilde{H}=K$ on $\partial X$. Set $H=\tilde{H}e^{f}$, it follows

(3.8)
$\begin{split}D_{H}^{\ast}\psi_{H}&=-\frac{1}{2}D_{\tilde{H}}^{\ast}(df\otimes\operatorname{id}_{E})=\frac{\operatorname{tr}D_{H}^{\ast}\psi_{H}}{\operatorname{rank}(E)}\operatorname{id}_{E}.\end{split}$

Now we set $f=\frac{\log\det(\tilde{H}^{-1}K)}{\operatorname{rank}(E)}$, then
we have $H=K$ on $\partial X$ and $\det H=\det K$. ∎

On each $M_{s}$, the following Dirichlet problem is solvable,

(3.9)
$\begin{split}\left\\{\begin{array}[]{ll}D_{H_{s}}^{\ast}\psi_{H_{s}}=c_{s}\operatorname{id}_{E},\\\
H|_{\partial M_{s}}=K,\\\ \det
h_{s}=1,h_{s}=K^{-1}H_{s}.\end{array}\right.\end{split}$

Due to the elementary inequality
$\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\geq\log(\det
h_{s})^{\frac{1}{r}}=0$ and the boundary condition, we have

(3.10)
$\begin{split}\frac{\partial\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})}{\partial\vec{n}}\geq
0,\end{split}$

where $\vec{n}$ is the inward unit normal vector field at $\partial M_{s}$. We
extend $\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})$ to the
function
$\widetilde{\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})}$ on
whole $M$ by setting $0$ outside $M_{s}$. For any non-negative compactly
supported function $\varphi$, the Green’s formula yields

(3.11)
$\begin{split}\int_{M}\widetilde{\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})}\Delta\varphi\operatorname{dvol}_{g}&=\int_{M_{s}}\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\Delta\varphi\operatorname{dvol}_{g}\\\
&=\int_{M_{s}}\varphi\Delta\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\operatorname{dvol}_{g}\\\
&+\int_{\partial
M_{s}}\varphi\frac{\partial\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})}{\partial\vec{n}}\operatorname{dvol}_{\partial
M_{s}}\\\ &-\int_{\partial
M_{s}}\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\frac{\partial\varphi}{\partial\vec{n}}\operatorname{dvol}_{\partial
M_{s}}\\\
&\geq\int_{M_{s}}\varphi\Delta\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\operatorname{dvol}_{g}.\end{split}$

Now by Lemma 2.2, we see

(3.12)
$\begin{split}\Delta\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})&=\frac{2(D_{H_{s}}^{\ast}\psi_{H_{s}}-D_{K}^{\ast}\psi_{K},h_{s})+|h_{s}^{-\frac{1}{2}}\delta_{K}h_{s}|^{2}}{\operatorname{tr}h_{s}}-\frac{|\operatorname{tr}\delta_{K}h_{s}|^{2}}{(\operatorname{tr}h_{s})^{2}}\\\
&\geq\frac{2(D_{H_{s}}^{\ast}\psi_{H_{s}}-D_{K}^{\ast}\psi_{K},h_{s})}{\operatorname{tr}h_{s}}\\\
&=\frac{-2((D_{K}^{\ast}\psi_{K})^{\perp},h_{s})}{\operatorname{tr}h_{s}}+\frac{2((\operatorname{tr}D_{H_{s}}^{\ast}\psi_{H_{s}}-\operatorname{tr}D_{K}^{\ast}\psi_{K})\operatorname{id}_{E},h_{s})}{\operatorname{rank}(E)\operatorname{tr}h_{s}}\\\
&\geq-2|(D_{K}^{\ast}\psi_{K})^{\perp}|+\frac{\Delta\log\det
h_{s}}{\operatorname{rank}(E)}\\\ &\geq-2C\phi,\end{split}$

where we have used the following simple inequality

(3.13)
$\begin{split}\frac{|\operatorname{tr}\delta_{K}h_{s}|^{2}}{\operatorname{tr}h_{s}}\leq|h_{s}^{-\frac{1}{2}}\delta_{K}h_{s}|^{2}.\end{split}$

And therefore

(3.14)
$\begin{split}\int_{M}\widetilde{\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})}\Delta\varphi\operatorname{dvol}_{g}&\geq-2C\int_{M_{s}}\varphi\phi\operatorname{dvol}_{g}\geq-2C\int_{M}\varphi\phi\operatorname{dvol}_{g}.\end{split}$

That is,
$\widetilde{\Delta\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})}\geq-2C\phi$
as a distribution. So by Assumption 2.1 we get

(3.15)
$\begin{split}\max\limits_{M_{s}}\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\leq
C_{1}(1+\int_{M_{s}}\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\phi\operatorname{dvol}_{g}),\end{split}$

where $C_{1}$ depends on $C$. Combing this with $\det h_{s}=1$ yields

(3.16) $\begin{split}\max\limits_{M_{s}}|\log h_{s}|\leq
C_{2}(1+\int_{M_{s}}|\log h_{s}|\phi\operatorname{dvol}_{g}),\end{split}$

for a constant $C_{2}$ depending only on $C_{1}$ and $\operatorname{rank}(E)$,
where we have used that

(3.17)
$\begin{split}\log(\frac{\operatorname{tr}h_{s}}{\operatorname{rank}(E)})\leq|\log
h_{s}|\leq\operatorname{rank}(E)^{\frac{3}{2}}\log\operatorname{tr}h_{s}.\end{split}$

Our goal is to show the quantity

(3.18) $\begin{split}\int_{M_{s}}|\log
h_{s}|\phi\operatorname{dvol}_{g}\end{split}$

is uniformly bounded, under the assumption that $D$ is $K$-analytically
stable.

For further consideration, we recall some notations. Given $\rho\in
C^{\infty}(\mathbb{R},\mathbb{R}),\Theta\in
C^{\infty}(\mathbb{R}\times\mathbb{R},\mathbb{R})$,
$\chi\in\Omega^{\bullet}(\operatorname{End}(E))$ and a self-adjoint bundle
endomorphism $\sigma$, we define $\rho[\sigma]$ and $\Theta[\sigma](\chi)$ as
follows. For every $x\in M$, we set

(3.19) $\begin{split}\rho[\sigma]=\rho(\lambda_{\alpha})e^{\alpha}\otimes
e_{\alpha},\Theta[\sigma](\chi)=\Theta(\lambda_{\alpha},\lambda_{\beta})\chi_{\alpha}^{\beta}e^{\alpha}\otimes
e_{\beta},\end{split}$

where $\\{e_{\alpha}\\}_{\alpha=1}^{\operatorname{rank}(E)}$ is an orthogonal
basis with respect to $K$, such that

(3.20)
$\begin{split}\sigma(e_{\alpha})=\lambda_{\alpha}e_{\alpha},\chi=\chi_{\alpha}^{\beta}e^{\alpha}\otimes
e_{\beta}.\end{split}$

Next we prove the following proposition.

######  Proposition 3.5.

Let $(E,D)$ be a vector bundle over a compact Riemannian manifold $(X,g)$ with
empty boundary, and $H,K$ be two fiber metrics with $s=\log(K^{-1}H)$ and
$s|_{\partial X}=0$. Then we have

(3.21)
$\begin{split}\int_{X}(D_{H}^{\ast}\psi_{H}-D_{K}^{\ast}\psi_{K},s)\operatorname{dvol}_{g}=-\frac{1}{2}\int_{X}(\Theta[s](Ds),Ds)\operatorname{dvol}_{g},\end{split}$

in the sense of $(\ref{definition})$, where

(3.22) $\begin{split}\Theta(x,y)=\begin{cases}\frac{e^{y-x}-1}{y-x},x\neq
y,\\\ 1,x=y.\end{cases}\end{split}$

###### Proof.

Consider the local normal coordinate at the considered point and for
$h=K^{-1}H$, noting that $(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,s)=\operatorname{tr}(s\delta_{K,\frac{\partial}{\partial x^{i}}}s)$,
through computing, we have

(3.23)
$\begin{split}(D_{H}^{\ast}\psi_{H}-D_{K}^{\ast}\psi_{K},s)&=-\frac{1}{2}(D_{K}^{\ast}(h^{-1}\delta_{K}h),s)-\frac{1}{2}([h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,\psi_{K}(\frac{\partial}{\partial x^{i}})],s)\\\
&=\frac{1}{2}(D_{K,\frac{\partial}{\partial
x^{i}}}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h),s)+\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,[\psi_{K}(\frac{\partial}{\partial x^{i}}),s])\\\
&=\frac{1}{2}\frac{\partial}{\partial
x^{i}}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,s)-\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,D_{K,\frac{\partial}{\partial x^{i}}}s)\\\
&+\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,[\psi_{K}(\frac{\partial}{\partial x^{i}}),s])\\\
&=\frac{1}{2}\frac{\partial}{\partial
x^{i}}\operatorname{tr}(s\delta_{K,\frac{\partial}{\partial
x^{i}}}s)-\frac{1}{2}(h^{-1}\delta_{K,\frac{\partial}{\partial
x^{i}}}h,\delta_{K,\frac{\partial}{\partial x^{i}}}s)\\\
&=\frac{1}{2}\frac{\partial}{\partial
x^{i}}\operatorname{tr}(s\frac{\partial}{\partial
x^{i}}s)-\frac{1}{2}(h^{-1}\delta_{K}h,\delta_{K}s).\end{split}$

On the other hand, according to Section 7.4 in [28], there is an open dense
subset $W\subset X$ and for every $x\in W$, we may choose an orthogonal basis
$\\{e_{\alpha}\\}_{\alpha=1}^{\operatorname{rank}(E)}$ for $E$ with respect to
$K$, defined over a neighborhood of $x$, such that

(3.24)
$\begin{split}h=\sum\limits_{\alpha=1}^{\operatorname{rank}(E)}e^{\lambda_{\alpha}}e_{\alpha}\otimes
e^{\alpha},s=\sum\limits_{\alpha=1}^{\operatorname{rank}(E)}\lambda_{\alpha}e_{\alpha}\otimes
e^{\alpha}.\end{split}$

We set $De_{\alpha}=A_{\alpha}^{\beta}e_{\beta}$, then it follows

(3.25) $\begin{split}Dh\circ
h^{-1}=\Theta[s](Ds)=\sum\limits_{\alpha=1}^{\operatorname{rank}(E)}d\lambda_{\alpha}e_{\alpha}\otimes
e^{\alpha}+\sum\limits_{\alpha\neq\beta}(1-e^{\lambda_{\alpha}-\lambda_{\beta}})A_{\beta}^{\alpha}e_{\alpha}\otimes
e^{\beta},\end{split}$

and therefore

(3.26) $\begin{split}(h^{-1}\delta_{K}h,\delta_{K}s)&=(Ds,Dh\circ
h^{-1})=(\Theta[s](Ds),Ds).\end{split}$

Finally, by $(\ref{key01})$ and $(\ref{key02})$, we have

(3.27)
$\begin{split}\int_{X}(D_{H}^{\ast}\psi_{H}-D_{K}^{\ast}\psi_{K},s)\operatorname{dvol}_{g}&=\frac{1}{4}\int_{X}d(\ast
d|s|^{2})-\frac{1}{2}\int_{X}(\Theta[s](Ds),Ds)\operatorname{dvol}_{g}\\\
&=-\frac{1}{2}\int_{X}(\Theta[s](Ds),Ds)\operatorname{dvol}_{g}.\end{split}$

∎

We shall prove $(\ref{sec32})$ by contradictory argument and the method is a
combination of Proposition 3.5 and Simpson’s trick. Otherwise, we may assume
that there exists a sequence $s\rightarrow\infty$, such that

(3.28) $\begin{split}l_{s}=\int_{M_{s}}|\log
h_{s}|\phi\operatorname{dvol}_{g}\rightarrow+\infty.\end{split}$

Let’s set $u_{s}=\log h_{s}/l_{s}$ with $\operatorname{tr}u_{s}=0$ and
$\int_{M_{s}}|u_{s}|\phi\operatorname{dvol}_{g}=1$. By $(\ref{sec31})$ we see

(3.29)
$\begin{split}\sup\limits_{M_{s}}|u_{s}|\leq\frac{C_{2}}{l_{s}}(1+l_{s})\leq
C_{3},\end{split}$

for a constant $C_{3}$ doesn’t depend on $s$. Using Proposition 3.5 and the
fact that $H_{s}$ is a Poisson metric, one has

(3.30)
$\begin{split}\int_{M_{s}}l_{s}(\Theta[l_{s}u_{s}](Du_{s}),Du_{s})\operatorname{dvol}_{g}&=2\int_{M_{s}}(D_{K}^{\ast}\psi_{K}-D_{H_{s}}^{\ast}\psi_{H_{s}},u_{s})\operatorname{dvol}_{g}\\\
&=2\int_{M_{s}}((D_{K}^{\ast}\psi_{K})^{\perp},u_{s})\operatorname{dvol}_{g}.\end{split}$

Notice that

(3.31)
$\begin{split}l\Theta(lx,ly)\rightarrow\begin{cases}\frac{1}{x-y},x>y,\\\
+\infty,x\leq y,\end{cases}\end{split}$

increases monotonically as $l\rightarrow+\infty$. It holds that

(3.32)
$\begin{split}\int_{M_{s}}(\rho[u_{s}](Du_{s}),Du_{s})\operatorname{dvol}_{g}\leq
2\int_{M_{s}}((D_{K}^{\ast}\psi_{K})^{\perp},u_{s})\operatorname{dvol}_{g}.\end{split}$

for any $\rho:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ with
$\rho(x,y)<\frac{1}{x-y}$ whenever $x>y$ and $s$ large enough.

Due to the zeroth order estimate of $u_{s}$, and
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$, by taking $\rho$ small enough,
it follows that $Du_{s}$ is bounded in $L^{2}$-norm on any compact subset of
$M$. We may assume that $u_{s}\rightarrow u_{\infty}$ weakly in
$L_{1,\operatorname{loc}}^{2}$-topology with $||u_{\infty}||_{L^{\infty}}\leq
C_{3}$. Since $\phi\in L^{1}$, for any $\epsilon>0$ there exists a
$s_{\epsilon}$ such that for any $s\geq s_{\epsilon}$, it holds

(3.33)
$\begin{split}\int_{M-M_{s}}\phi\operatorname{dvol}_{g}\leq\epsilon.\end{split}$

Hence for $s_{\epsilon}\leq s_{1}\leq s_{2}$, $(\ref{sec33})$ indicates

(3.34)
$\begin{split}1-C_{3}\epsilon&\leq\int_{M_{s_{2}}}|u_{s_{2}}|\phi\operatorname{dvol}_{g}-\int_{M_{s_{2}}-M_{s_{1}}}|u_{s_{2}}|\phi\operatorname{dvol}_{g}=\int_{M_{s_{1}}}|u_{s_{2}}|\phi\operatorname{dvol}_{g}\leq
1.\end{split}$

Fixing $s_{1}$ and by the weak compactness of $\\{u_{s}\\}$ in
$L_{1,\operatorname{loc}}^{2}$-topology, we get

(3.35)
$\begin{split}1-C_{3}\epsilon\leq\int_{M_{s_{1}}}|u_{\infty}|\phi\operatorname{dvol}_{g}\leq
1,\end{split}$

and then taking $s_{1}\rightarrow\infty$ and $\epsilon\rightarrow 0$ yields
$\int_{M}|u_{\infty}|\phi\operatorname{dvol}_{g}=1$, which implies
$u_{\infty}\neq 0$.

Now it follows from $(\ref{sec33}),(\ref{sec34}),(\ref{sec35})$ and
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ that

(3.36)
$\begin{split}\int_{M_{s_{1}}}(\rho[u_{s_{2}}](Du_{s_{2}}),Du_{s_{2}})\operatorname{dvol}_{g}&\leq\int_{M_{s_{2}}}(\rho[u_{s_{2}}](Du_{s_{2}}),Du_{s_{2}})\operatorname{dvol}_{g}\\\
&\leq
2(\int_{M_{s_{2}}-M_{s_{1}}}+\int_{M_{s_{1}}})((D_{K}^{\ast}\psi_{K})^{\perp},u_{s_{2}})\operatorname{dvol}_{g}\\\
&\leq
2CC_{3}\epsilon+2\int_{M_{s_{1}}}((D_{K}^{\ast}\psi_{K})^{\perp},u_{s_{2}})\operatorname{dvol}_{g}.\end{split}$

Let’s take $s_{2}\rightarrow\infty,s_{1}\rightarrow\infty$ and
$\epsilon\rightarrow 0$ successively, it yields

(3.37)
$\begin{split}\int_{M}(\rho[u_{\infty}](Du_{\infty}),Du_{\infty})\operatorname{dvol}_{g}&\leq
2\int_{M}((D_{K}^{\ast}\psi_{K})^{\perp},u_{\infty})\operatorname{dvol}_{g}\leq
2CC_{3}\int_{M}\phi\operatorname{dvol}_{g}.\end{split}$

As long as $(\ref{sec36})$ is established, by using Simpson’s discussion in
[39](p886-888) and Loftin’s observation in [27], it’s easy to conclude
$(\ref{sec32})$ and therefore we obtain

######  Proposition 3.6.

If $(M,g)$ satisfies the Assumption 2.1 and the flat bundle $(E,D)$ is
$K$-analytically stable with $|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for
a constant $C$, it holds

(3.38) $\begin{split}\max\limits_{M_{s}}|\log
h_{s}|\leq\tilde{C}_{1},\end{split}$

for a constant $\tilde{C}_{1}$ that doesn’t depend on $s$.

Using Lemma 2.2 again shows

(3.39)
$\begin{split}|h_{s}^{-\frac{1}{2}}\delta_{K}h_{s}||_{L^{2}(M_{s})}^{2}&=\int_{M_{s}}\left(\Delta\operatorname{tr}h_{s}-2(D_{H_{s}}^{\ast}\psi_{H_{s}}-D_{K}^{\ast}\psi_{K},h_{s})\right)\operatorname{dvol}_{g}\\\
&=\int_{M_{s}}\left(\Delta\operatorname{tr}h_{s}+2((D_{K}^{\ast}\psi_{K})^{\perp},h_{s})\right)\operatorname{dvol}_{g}\\\
&=-\int_{\partial
M_{s}}\frac{\partial\operatorname{tr}h_{s}}{\partial\vec{n}}\operatorname{dvol}_{\partial
M_{s}}+2\int_{M_{s}}((D_{K}^{\ast}\psi_{K})^{\perp},h_{s})\operatorname{dvol}_{g}\\\
&\leq
2\int_{M_{s}}((D_{K}^{\ast}\psi_{K})^{\perp},h_{s})\operatorname{dvol}_{g}\\\
&\leq CC_{4}\int_{M}\phi\operatorname{dvol}_{g},\end{split}$

where $C_{4}$ depends only on $\tilde{C}_{1}$ and $\operatorname{rank}(E)$. So
we obtain

######  Proposition 3.7.

If $(M,g)$ satisfies the Assumption 2.1 and the flat bundle $(E,D)$ is
$K$-analytically stable with $|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for
a constant $C$, we have

(3.40)
$\begin{split}||Dh_{s}||_{L^{2}(M_{s})}\leq\tilde{C_{2}}||\phi||_{L^{1}}^{\frac{1}{2}},\end{split}$

for a constant $\tilde{C}_{2}$ that depends only on $C,\tilde{C}_{1}$ and
$\operatorname{rank}(E)$.

######  Proposition 3.8.

If $(M,g)$ satisfies the Assumption 2.1 and the flat bundle $(E,D)$ is
$K$-analytically stable with $|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for
a constant $C$, then there exists a Poisson metric $H$ such that $\det h=1$,
$|h|\in L^{\infty}$ and $|Dh|\in L^{2}$ for $h=K^{-1}H$.

###### Proof.

Using the flatness of $D$, we deduce

(3.41)
$\begin{split}\Delta|\psi_{H_{s}}|^{2}_{H_{s}}&\geq-2g^{ij}(D_{H_{s}}^{2}(\cdot,\frac{\partial}{\partial
x^{i}})(\psi_{H_{s}}(\frac{\partial}{\partial
x^{j}})),\psi_{H_{s}})_{H_{s}}\\\
&+2g^{ij}(\psi_{H_{s}}(R_{\nabla}(\cdot,\frac{\partial}{\partial
x^{i}})(\frac{\partial}{\partial
x^{j}})),\psi_{H_{s}})_{H_{s}}-2(D_{H_{s}}D_{H_{s}}^{\ast}\psi_{H_{s}},\psi_{H_{s}})_{H_{s}}\\\
&=|[\psi_{H_{s}},\psi_{H_{s}}]|_{H_{s}}^{2}+2(\psi_{H_{s}}\circ\operatorname{Ric},\psi_{H_{s}})-2(D_{H_{s}}D_{H_{s}}^{\ast}\psi_{H_{s}},\psi_{H_{s}})_{H_{s}}\\\
&\geq-
C_{5}|\psi_{H_{s}}|_{H_{s}}^{2}-\frac{2}{\operatorname{rank}(E)}(d\operatorname{tr}D_{H_{s}}^{\ast}\psi_{H_{s}}\otimes\operatorname{id}_{E},\psi_{H_{s}})_{H_{s}}\\\
&=-C_{5}|\psi_{H_{s}}|_{H_{s}}^{2}-\frac{2}{\operatorname{rank}(E)}(d\operatorname{tr}D_{K}^{\ast}\psi_{K}\otimes\operatorname{id}_{E},\psi_{H_{s}})_{H_{s}}\\\
&\geq-C_{6}|\psi_{H_{s}}|_{H_{s}}^{2}-C_{7},\end{split}$

where $C_{6}$ depends only on the lower bounded of the Ricci curvature at the
considered point, $C_{7}$ depends only on
$|d\operatorname{tr}D_{K}^{\ast}\psi_{K}|$ and $\operatorname{rank}(E)$.

On the other hand, $(\ref{psiHpsiK})$ implies there exists two constants
$C_{8},C_{9}$ that depend only on $\tilde{C}_{1}$ and
$\operatorname{rank}(E)$, such that

(3.42) $\begin{split}|\psi_{H_{s}}|_{H_{s}}^{2}\leq
C_{8}(|\psi_{K}|^{2}+|Dh_{s}|^{2}),\end{split}$ (3.43)
$\begin{split}|Dh_{s}|^{2}\leq
C_{9}(|\psi_{K}|^{2}+|\psi_{H_{s}}|^{2}_{H_{s}}).\end{split}$

So combing $(\ref{sec38})$, $(\ref{sec39})$ with Proposition 3.7,
$(\ref{sec37})$ and the Theorem 9.20 in [18], we get the uniform local
boundedness of $|Dh_{s}|$. Now on each $M_{s}$, by the Poisson metric
equation, we have

(3.44)
$\begin{split}D_{K}^{\ast}(h_{s}^{-1}\delta_{K}h_{s})=2(D_{K}^{\ast}\psi_{K})^{\perp}-g^{ij}[h^{-1}_{s}\delta_{K,\frac{\partial}{\partial
x^{i}}}h_{s},\psi_{K}(\frac{\partial}{\partial x^{j}})],\end{split}$

the standard bootstrapping procedure implies the uniform local higher order
estimates of $\\{h_{s}\\}$ and by diagonal subsequence argument, $H_{s}$
converge to a metric $H_{\infty}$ on the whole $M$ in
$C_{\operatorname{loc}}^{\infty}$-topology. It obvious that $H_{\infty}$ is a
Poisson metric on $(E,D)$ such that $\det h=1$, $|h|\in L^{\infty}$ and
$|Dh|\in L^{2}$. ∎

######  Proof of Theorem 1.3.

The existence is given by Proposition 3.8 and the uniqueness will be proved in
Proposition 4.1. Conversely, we assume $D$ is irreducible and $H$ be a Poisson
metric which shares the same properties with $H_{\infty}$, we shall
demonstrate that it is also analytically stable with respect to $K$. For any
$D$-invariant sub-bundle $S\subset E$, $K$ and $H$ restrict to the metrics
$K_{S}$ and $H_{S}$ on $S$. Taking the orthogonal complement of $S$ in $E$
with respect to $K$, we have the orthogonal decomposition $E=S\oplus
S^{\perp}$ and the projection $\pi_{K}$ onto $S$.

If $|D\pi_{K}|\not\in L^{2}$, by $(\ref{Chernweilbundle})$ and
$|D_{K}^{\ast}\psi_{K}|\in L^{1}$, we know

(3.45) $\begin{split}\deg_{g}(S,D,K)=-\infty.\end{split}$

If $|D\pi_{K}|\in L^{2}$, we know $\deg_{g}(S,D,K)$ is finite. Moreover, we
have

(3.46) $\begin{split}h_{S}=\pi_{K}\circ h\circ\pi_{K},\end{split}$ (3.47)
$\begin{split}D_{S}h_{S}=\pi_{K}\circ
Dh\circ\pi_{K}+D\pi_{K}\circ(\operatorname{id}_{E}-\pi_{K})\circ
h\circ\pi_{K},\end{split}$

where $h_{S}=K_{S}^{-1}H_{S}$ and $D_{S}$ is the induced connection on $S$. It
then follows

(3.48)
$\begin{split}\int_{M}|\operatorname{tr}(h_{S}^{-1}\delta_{D_{S},K_{S}}h_{S})|^{2}\operatorname{dvol}_{g}&\leq\int_{M}|h_{S}^{-1}|^{2}|D_{S}h_{S}|^{2}\operatorname{dvol}_{g}<\infty,\end{split}$

where $\delta_{D_{S},K_{S}}=D_{S,K}-\psi_{D_{S},K_{S}}$. So Simpson’s Lemma
5.2 in [39](or Yau’s Lemma in [46]) implies

(3.49)
$\begin{split}\lim\limits_{j\rightarrow\infty}\int_{M_{j}}\frac{1}{2}d(\ast
d\log\det
h_{S})=\lim\limits_{j\rightarrow\infty}\int_{M_{j}}\frac{1}{2}d(\ast\operatorname{tr}(h_{S}^{-1}\delta_{D_{S},K_{S}}h_{S}))=0,\end{split}$

where each $M_{j}$ is an exhaustion subset. It then follows from the
convergence theorem that

(3.50)
$\begin{split}\lim\limits_{j\rightarrow\infty}\int_{M_{j}}\operatorname{tr}D_{H_{S}}^{\ast}\psi_{H_{S}}\operatorname{dvol}_{g}&=\lim\limits_{j\rightarrow\infty}\int_{M_{j}}\operatorname{tr}D_{K_{S}}^{\ast}\psi_{K_{S}}\operatorname{dvol}_{g}\\\
&+\lim\limits_{j\rightarrow\infty}\int_{M_{j}}\frac{1}{2}\Delta\log\det
h_{S}\operatorname{dvol}_{g}\\\ &=-\deg_{g}(S,D,K),\end{split}$

and hence

(3.51)
$\begin{split}\lim\limits_{j\rightarrow\infty}\int_{M_{j}}\operatorname{tr}D_{H_{S}}^{\ast}\psi_{H_{S}}\operatorname{dvol}_{g}&=\int_{M}\operatorname{tr}D_{H_{S}}^{\ast}\psi_{H_{S}}\operatorname{dvol}_{g}=-\deg_{g}(S,D,H).\end{split}$

In summary, we conclude either

(3.52) $\begin{split}\deg_{g}(S,D,K)=-\infty\end{split}$

or

(3.53) $\begin{split}\deg_{g}(S,D,K)=\deg_{g}(S,D,H).\end{split}$

On the other hand, it follows from Proposition 2.1 that $E$ is
$H$-analytically stable and hence the proof is completed. ∎

##  4\. Some applications and consequences

###  4.1. Uniqueness of Poisson metrics

Firstly, we get the following uniqueness result, which implies the limiting
Poisson metric in Theorem 1.3 is unique.

######  Proposition 4.1.

Let $(E,D)$ be a $K$-analytically stable bundle over a Riemannian manifold
$(M,g)$ satisfying the Assumptions 2.1 and either Assumption 2.2 or complete
with finite volume. Suppose $\operatorname{tr}D_{K}^{\ast}\psi_{K}\in L^{1}$
and $H$ be a Poisson metric such that $\det h=1,|h|\in L^{\infty}$ and
$|Dh|\in L^{2}$, where $h=K^{-1}H$. If $\tilde{H}$ is another Poisson metric
which is mutually bounded with $H$ and $\det H=\det\tilde{H}$, we have
$H=\tilde{H}$.

###### Proof.

Let $\tilde{h}=H^{-1}\tilde{H}$ and then we have

(4.1)
$\begin{split}\Delta\operatorname{tr}\tilde{h}&=2(D_{\tilde{H}}^{\ast}\psi_{\tilde{H}}-D_{H}^{\ast}\psi_{H},\tilde{h})_{H}+|\tilde{h}^{-\frac{1}{2}}\delta_{H}\tilde{h}|^{2}_{H}\\\
&=\frac{(\Delta\log\det\tilde{h}\operatorname{id}_{E},\tilde{h})_{H}}{\operatorname{rank}(E)}+|\tilde{h}^{-\frac{1}{2}}\delta_{H}\tilde{h}|^{2}_{H}\\\
&=|\tilde{h}^{-\frac{1}{2}}\delta_{H}\tilde{h}|^{2}_{H}.\end{split}$

So Assumption 2.1 implies $\tilde{h}$ is $D$-parallel and set
$E=\mathop{\bigoplus}\limits_{j=1}^{m}E_{j}$ be the eigendecomposition of
$\tilde{h}$, then this decomposition is orthogonal with respect $H$ and
$\tilde{H}$. Moreover, we have $\tilde{H}|_{E_{j}}=c_{j}H|_{E_{j}}$ for some
constants $c_{j}$ and

(4.2)
$\begin{split}-\int_{M}\operatorname{tr}D_{H}^{\ast}\psi_{H}\operatorname{dvol}_{g}=-\sum\limits_{j=1}^{m}\int_{M}\operatorname{tr}D_{H|_{E_{j}}}^{\ast}\psi_{H|_{E_{j}}}\operatorname{dvol}_{g}.\end{split}$

Then running the same argument in the proof of Theorem 1.3 shows that $D$
being $H$-analytically stable. As a consequence,

(4.3)
$\begin{split}\frac{\deg(E,D,H)}{\operatorname{rank}(E)}&=-\sum\limits_{j=1}^{m}\frac{\int_{M}\operatorname{tr}D_{H|_{E_{j}}}^{\ast}\psi_{H|_{E_{j}}}\operatorname{dvol}_{g}}{\operatorname{rank}(E_{j})}\frac{\operatorname{rank}(E_{j})}{\operatorname{rank}(E)}\\\
&\leq\frac{\deg(E,D,H)}{\operatorname{rank}(E)},\end{split}$

the strict inequality occurs if and only if $m>1$ and therefore we complete
the proof. ∎

If $D$ being flat, the following is a flat bundle analogy of Mochizuki’s
uniqueness result on Hermitian-Einstein metrics in [33].

######  Proposition 4.2.

Let $(E,D)$ be a $K$-analytically stable flat bundle over a Riemannian
manifold $(M,g)$ satisfying Assumption 2.1 and there is a positive exhaustion
function $\rho$ such that $|d\log\rho|\in L^{1}$. Assume that $|\psi_{K}|\in
L^{2}$ and $H_{1}$ be a Poisson metric such that $|h_{1}|,|h_{1}^{-1}|\in
L^{\infty}$ and $|Dh_{1}|\in L^{2}$, where $h_{1}=K^{-1}H_{1}$. If $H_{2}$ is
another Poisson metric which is mutually bounded with $H_{1}$ and $\det
H_{1}=\det H_{2}$, then $H_{1}=H_{2}$.

###### Proof.

Similar, set $E=\mathop{\bigoplus}\limits_{j=1}^{m}E_{j}$ be the
eigendecomposition of $h_{12}=H_{1}^{-1}H_{2}$ and
$H_{1}|_{E_{j}}=c_{i}H_{2}|_{E_{j}}$ for some constants $c_{j}$. Let $\pi_{j}$
denotes the projection onto $E_{j}$ and $h_{i}=K^{-1}H_{i}$, then $\pi_{j}$
are bounded with respect to $K$, as $K$ and $H_{i}$ are mutually bounded.

Now $(\ref{deltaHdeltaK})$ indicates
$\delta_{K}\pi_{j}=\delta_{H_{1}}\pi_{j}-[h_{1}^{-1}\delta_{K}h_{1},\pi_{j}]=-[h_{1}^{-1}\delta_{K}h_{1},\pi_{j}]$,
this implies $\delta_{K}\pi_{j}$ and $D\pi_{j}^{\ast K}$ are $L^{2}$. We
consider the fiber metric $H$ determined by the direct sum of $K|_{E_{j}}$,
then $H$ and $K$ are mutually bounded. By definition, we have
$h:=K^{-1}H=\sum\limits_{j=1}^{m}\pi_{j}^{\ast K}\circ\pi_{j}$ and it follows

(4.4) $\begin{split}h^{-1}Dh=h^{-1}\sum\limits_{j=1}^{m}D\pi_{j}^{\ast
K}\circ\pi_{j},h^{-1}\delta_{K}h=h^{-1}\sum\limits_{j=1}^{m}\pi_{j}^{\ast
K}\circ\delta_{K}\pi_{j},\end{split}$

which implies $h^{-1}Dh$ and $h^{-1}\delta_{K}h$ are $L^{2}$.

Next we compute
$D(h^{-1}\delta_{K}h)=-h^{-1}Dhh^{-1}\delta_{K}h+h^{-1}D(\delta_{K}h)$ and

(4.5) $\begin{split}D(\delta_{K}h)&=D(\sum\limits_{j=1}^{m}\pi_{j}^{\ast
K}\circ\delta_{K}\pi_{j})\\\ &=\sum\limits_{j=1}^{m}\left(D\pi_{j}^{\ast
K}\circ\delta_{K}\pi_{j}+\pi_{j}^{\ast K}\circ D(\delta_{K}\pi_{j})\right)\\\
&=\sum\limits_{j=1}^{m}\left(D\pi_{j}^{\ast
K}\circ\delta_{K}\pi_{j}+4\pi_{j}^{\ast K}\circ[D_{K}^{2},\pi_{j}]\right)\\\
&=\sum\limits_{j=1}^{m}\left(D\pi_{j}^{\ast
K}\circ\delta_{K}\pi_{j}-2\pi_{j}^{\ast
K}\circ[[\psi_{K},\psi_{K}],\pi_{j}]\right).\end{split}$

where we have use the flatness of $D$. As a consequence, we find
$\Delta\log\det h$ is $L^{1}$.

Finally we set $\chi_{N}=\eta(\frac{\rho}{N})$, where $\eta$ is a nonnegative
function with $\eta(t)=0$ if $t\geq 2$ and $\eta(t)=1$ if $t\leq 1$. We then
deduce

(4.6) $\begin{split}\int_{M}\chi_{N}\Delta\log\det
h\operatorname{dvol}_{g}&=-\int_{M}\nabla\chi_{N}\cdot\nabla\log\det
h\operatorname{dvol}_{g}\\\ &=-\int_{\\{N\leq\rho\leq
2N\\}}\frac{\eta^{{}^{\prime}}(\frac{\rho}{N})}{N}\nabla\rho\cdot\nabla\log\det
h\operatorname{dvol}_{g}\\\ &\leq\int_{\\{N\leq\rho\leq
2N\\}}\frac{2|\eta^{{}^{\prime}}(\frac{\rho}{N})|}{\rho}|\nabla\rho||\nabla\log\det
h|\operatorname{dvol}_{g}\\\ &=\int_{\\{N\leq\rho\leq
2N\\}}2|\eta^{{}^{\prime}}(\frac{\rho}{N})||\nabla\log\rho||\operatorname{tr}(h^{-1}\delta_{K}h)|\operatorname{dvol}_{g}.\end{split}$

Combing this and using the fact that $\Delta\log\det h$ lies in $L^{1}$, it
follows

(4.7)
$\begin{split}-\int_{M}\operatorname{tr}D_{K}^{\ast}\psi_{K}\operatorname{dvol}_{g}&=-\int_{M}\operatorname{tr}D_{H}^{\ast}\psi_{H}\operatorname{dvol}_{g}+\frac{1}{2}\int_{M}\Delta\log\det
h\operatorname{dvol}_{g}\\\
&=-\sum\limits_{j=1}^{m}\int_{M}\operatorname{tr}D_{K|_{E_{j}}}^{\ast}\psi_{K|_{E_{j}}}\operatorname{dvol}_{g}.\end{split}$

Since $D$ is $K$-analytically stable, we obtain $m=1$ and the proof is
completed. ∎

###  4.2. Applications to the nonableian Hodge correspondence

######  Proposition 4.3.

Let $(X,\omega)$ be a noncompact complex curve satisfying the Assumption 2.1,
and $(E,D)$ be a $K$-analytically stable flat bundle over $X$ with
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for a constant $C$, then there
exists a Higgs structure on $E$.

###### Proof.

According to Theorem 1.3, there is a Poisson metric $H$ on $(E,D)$ and we have

(4.8)
$\begin{split}D_{H}\psi_{H}^{\perp}=D_{H}(\psi_{H}^{\perp,1,0}+\psi_{H}^{\perp,0,1})=0,\end{split}$
(4.9) $\begin{split}D_{H}^{\ast}\psi_{H}^{\perp}=\sqrt{-1}\ast
D_{H}(\psi_{H}^{\perp,1,0}-\psi_{H}^{\perp,0,1})=0,\end{split}$

where $\psi_{H}^{\perp}$ is the trace-free part of $\psi_{H}$. Then we easily
get $D_{H}^{0,1}\psi_{H}^{\perp,1,0}=0$ and the pair
$(D_{H}^{0,1},\psi_{H}^{\perp,1,0})$ gives rise a Higgs structure on $E$. ∎

For higher dimensional case, we have

######  Proposition 4.4.

Let $(X,\omega)$ be a complete Kähler manifold with bounded Ricci curvature
from below and satisfying the Assumption 2.1. Assume $(E,D)$ be a
$K$-analytically stable flat bundle over $X$ with
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for a constant $C$ and
$|\psi_{K}|\in L^{2}$. Then there exists a Higgs structure on $E$.

###### Proof.

Due to Theorem 1.3, there is a Poisson metric $H$ and we compute

(4.10)
$\begin{split}\Delta|\psi_{H}^{\perp}|^{2}_{H}&=-2(\nabla_{H}^{\ast}\nabla_{H}\psi_{H}^{\perp},\psi_{H}^{\perp})_{H}+2|\nabla_{H}\psi_{H}^{\perp}|^{2}_{H}\\\
&=-2g^{ij}(D_{H}^{2}(\cdot,\frac{\partial}{\partial
x^{i}})(\psi_{H}^{\perp}(\frac{\partial}{\partial
x^{j}})),\psi_{H}^{\perp})_{H}\\\
&+2g^{ij}(\psi_{H}^{\perp}(R_{\nabla}(\cdot,\frac{\partial}{\partial
x^{i}})(\frac{\partial}{\partial
x^{j}})),\psi_{H}^{\perp})_{H}+2|\nabla_{H}\psi_{H}^{\perp}|_{H}^{2}\\\
&=g^{ij}([[\psi_{H},\psi_{H}(\frac{\partial}{\partial
x^{i}})],\psi_{H}^{\perp}(\frac{\partial}{\partial
x^{j}})]+2\psi_{H}^{\perp}\circ\operatorname{Ric},\psi_{H}^{\perp})_{H}+2|\nabla_{H}\psi_{H}^{\perp}|_{H}^{2}\\\
&=|[\psi_{H}^{\perp},\psi_{H}^{\perp}]|_{H}^{2}+2(\psi_{H}^{\perp}\circ\operatorname{Ric},\psi_{H}^{\perp})_{H}+2|\nabla_{H}\psi_{H}^{\perp}|^{2}_{H}\\\
&\geq-
C_{\ast}|\psi_{H}^{\perp}|_{H}^{2}+2|\nabla_{H}\psi_{H}^{\perp}|^{2}_{H},\end{split}$

where $-C_{\ast}$ the lower bound of Ricci curvature. Let $R$ be any positive
constant and we fix point $x_{0}$, choose a cut-off function $\eta$ satisfying

(4.11) $\begin{split}\left\\{\begin{array}[]{ll}\eta(x)=1,x\in
B_{x_{0}}(R),\\\ \eta(x)=0,x\in X\setminus B_{x_{0}}(2R),\\\ 0\leq\eta\leq
1,|\nabla\eta|\leq\frac{C_{1}}{R},\end{array}\right.\end{split}$

where $C_{1}$ is a positive constant and $B_{x_{0}}(R)$ is the geodesics ball
centered at $x_{0}$ with radius $R$. Then we have

(4.12)
$\begin{split}2\int_{X}\eta^{2}|\nabla_{H}\psi_{H}^{\perp}|^{2}_{H}\operatorname{dvol}_{\omega}&\leq\int_{X}\eta^{2}(\Delta|\psi_{H}^{\perp}|^{2}_{H}+C_{\ast}|\psi_{H}^{\perp}|^{2}_{H})\operatorname{dvol}_{\omega}\\\
&\leq\int_{X}4\eta|\nabla\eta||\psi_{H}^{\perp}|_{H}|\nabla_{H}\psi_{H}^{\perp}|_{H}\operatorname{dvol}_{\omega}+C_{\ast}\int_{X}\eta^{2}|\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}\\\
&\leq\int_{X}\eta^{2}|\nabla_{H}\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}+\int_{X}(C_{\ast}\eta^{2}+4|\nabla\eta|^{2})|\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}\\\
&\leq\int_{X}\eta^{2}|\nabla_{H}\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}+\int_{X}(C_{\ast}\eta^{2}+\frac{4C_{1}^{2}}{R^{2}})|\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}.\end{split}$

And it also holds

(4.13)
$\begin{split}\int_{X}|\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}&=\int_{X}|\psi_{K}^{\perp}-\frac{1}{2}h^{-1}Dh+h^{-1}[\psi_{K}^{\perp},h]|_{H}^{2}\operatorname{dvol}_{\omega}<\infty,\end{split}$

where $h=K^{-1}H$. So by letting $R$ goes infinity, we see

(4.14)
$\begin{split}\int_{X}|\nabla_{H}\psi_{H}^{\perp}|^{2}_{H}\operatorname{dvol}_{\omega}\leq
C_{\ast}\int_{X}|\psi_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega}<\infty.\end{split}$

Now let’s consider the pseudo-curvature operator
$G_{H}=(D_{H}^{0,1}+\psi_{H}^{1,0})^{2}$. Notice the flatness of $D$ implies

(4.15)
$\begin{split}D_{H}^{1,0}\psi_{H}^{0,1}=-D_{H}^{0,1}\psi_{H}^{1,0},D_{H}^{1,0}D_{H}^{1,0}=-\frac{1}{2}[\psi_{H}^{1,0},\psi_{H}^{1,0}],D_{H}^{0,1}D_{H}^{0,1}=-\frac{1}{2}[\psi_{H}^{0,1},\psi_{H}^{0,1}],\end{split}$

and it follows that

(4.16)
$\begin{split}G_{H}^{\ast}&=\left((D_{H}^{0,1})^{2}+D_{H}^{0,1}\psi_{H}^{1,0}+\psi_{H}^{1,0}\wedge\psi_{H}^{1,0}\right)^{\ast}\\\
&=-D_{H}^{1,0}D_{H}^{1,0}+D_{H}^{1,0}\psi_{H}^{0,1}-\psi_{H}^{0,1}\wedge\psi_{H}^{0,1}\\\
&=G^{2,0}_{H}-G_{H}^{1,1}+G^{0,2}_{H}.\end{split}$

By Kähler identity we know
$\sqrt{-1}\Lambda_{\omega}G_{H}^{\perp}=\frac{1}{2}(D_{H}^{\ast}\psi_{H})^{\perp}=0$
and

(4.17) $\begin{split}\operatorname{tr}(G_{H}^{\perp}\wedge
G_{H}^{\perp})\wedge\frac{\omega^{n-2}}{(n-2)!}&=\operatorname{tr}\left(G_{H}^{\perp}\wedge\ast(G_{H}^{\perp})^{\ast}\right)=|G_{H}^{\perp}|_{H}^{2}\operatorname{dvol}_{\omega},\end{split}$

where $n$ is the complex dimension of $X$. On the other hand, a direct
calculation shows

(4.18) $\begin{split}\operatorname{tr}(G_{H}^{\perp}\wedge
G_{H}^{\perp})\wedge\omega^{n-2}&=-2\operatorname{tr}(\psi_{H}^{0,1}\wedge\psi_{H}^{0,1}\wedge\psi_{H}^{1,0}\wedge\psi_{H}^{1,0})\wedge\omega^{n-2}\\\
&-\operatorname{tr}(D_{H}^{0,1}\psi_{H}^{1,0}\wedge
D_{H}^{1,0}\psi_{H}^{0,1})\wedge\omega^{n-2}\\\
&-\frac{\operatorname{tr}(D_{H}^{0,1}\psi_{H}^{1,0})\wedge\operatorname{tr}(D_{H}^{0,1}\psi_{H}^{1,0})}{\operatorname{rank}(E)}\wedge\omega^{n-2}\\\
&=-\overline{\partial}\left(\operatorname{tr}(\psi_{H}^{1,0}\wedge
D_{H}^{1,0}\psi_{H}^{0,1})\wedge\omega^{n-2}\right)\\\
&+\overline{\partial}\left(\frac{\operatorname{tr}\psi_{H}^{1,0}\wedge\operatorname{tr}(D_{H}^{1,0}\psi_{H}^{0,1})}{\operatorname{rank}(E)}\wedge\omega^{n-2}\right)\\\
&=-\overline{\partial}\left(\operatorname{tr}(\psi_{H}^{1,0}\wedge
D_{H}^{1,0}\psi_{H}^{\perp})\wedge\omega^{n-2}\right).\end{split}$

where we have used $(\ref{sec424})$, $D_{H}^{0,1}\psi_{H}^{0,1}=0$ and
$[D_{H}^{1,0},D_{H}^{0,1}]=-[\psi_{H}^{1,0},\psi_{H}^{0,1}]$.

Now as it holds

(4.19)
$\begin{split}D_{H}^{1,0}\psi_{H}^{\perp}=\sum\limits_{i=1}^{n}dz^{i}\wedge\nabla_{H,\frac{\partial}{\partial
z^{i}}}\psi_{H}^{\perp},\end{split}$

it follows from $(\ref{sec423})$ that

(4.20) $\begin{split}\int_{X}|\operatorname{tr}(\psi_{H}^{1,0}\wedge
D_{H}^{1,0}\psi_{H}^{\perp})\wedge\omega^{n-2}|\operatorname{dvol}_{\omega}<\infty.\end{split}$

Therefore, by $(\ref{sec425})$, $(\ref{sec426})$, $(\ref{sec427})$ and Yau’s
Lemma in [46], we conclude $G_{H}^{\perp}$ vanishes identically and the pair
$(D_{H}^{0,1},\psi_{H}^{\perp,1,0})$ is a Higgs structure on $E$. ∎

######  Proof of Theorem 1.4.

Proposition 4.3 and Proposition 4.4 indicate
$(D_{H}^{0,1},\psi_{H}^{\perp,1,0})$ determines a Higgs structure on $E$.
Moreover, we see the corresponding Hitchin-Simpson curvature

(4.21)
$\begin{split}F_{D_{H}^{0,1},\psi_{H}^{\perp,1,0},H}=D_{H}^{2}+D_{H}\psi_{H}^{\perp}+\frac{1}{2}[\psi_{H}^{\perp},\psi_{H}^{\perp}]=0,\end{split}$

where we have used the flatness of $D$. Hence we complete of proof of $(1)$
and $(2)$. For $(3)$, if $(X,\omega)$ has nonnegative Ricci curvature, we have
for any $\epsilon>0$ that

(4.22)
$\begin{split}\Delta(\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}+\epsilon)^{\frac{1}{2}}&=\frac{1}{4}\frac{\Delta|\psi_{H}^{\perp}|^{2}_{H}}{(\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}+\epsilon)^{\frac{1}{2}}}-\frac{1}{4}\frac{|\frac{1}{2}d|\psi_{H}^{\perp}|^{2}_{H}|^{2}}{(\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}+\epsilon)^{\frac{3}{2}}}\\\
&\geq\frac{1}{4}\frac{2|\nabla_{H}\psi_{H}^{\perp}|^{2}_{H}}{(\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}+\epsilon)^{\frac{1}{2}}}-\frac{1}{4}\frac{|\psi_{H}^{\perp}|^{2}_{H}|\nabla_{H}\psi_{H}^{\perp}|_{H}^{2}}{(\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}+\epsilon)^{\frac{3}{2}}}\\\
&=\frac{1}{2}\frac{|\nabla_{H}\psi_{H}^{\perp}|_{H}^{2}}{(\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}+\epsilon)^{\frac{1}{2}}}(1-\frac{\frac{1}{2}|\psi_{H}^{\perp}|^{2}_{H}}{\frac{1}{2}|\psi_{H}^{\perp}|_{H}^{2}+\epsilon})\\\
&\geq 0,\end{split}$

and by letting $\epsilon$ goes zero, it follows that $|\psi_{H}^{\perp}|_{H}$
is subharmonic. On the other hand, as $||\psi_{H}^{\perp}||_{H,L^{2}}<\infty$,
Yau’s Liouville theorem in [46] applies and we conclude
$|\psi_{H}^{\perp}|_{H}$ is constant. But the volume of $(X,\omega)$ is
infinite, so we know $\psi_{H}^{\perp}=0$ and then $(3)$ follows easily. ∎

In a similar way, using Corollary 1.2 instead, we obtain the following
proposition, which means the harmonic metric produced there is pluriharmonic.

######  Proposition 4.5.

Let $(X,\omega)$ be a Kähler manifold satisfying the Assumptions 2.1, 2.3 with
$\phi=1$ and $(E,D)$ be a $K$-analytically stable flat bundle over $X$ such
that $|D_{K}^{\ast}\psi_{K}|\in L^{\infty}$ and $|\psi_{K}|\in L^{2}$. Assume
either $\dim_{\mathbb{C}}X=1$ or $(X,\omega)$ being complete with bounded
Ricci curvature from below, then $(E,D)$ comes from a Higgs bundle.

Suppose $(E,D)$ be a flat bundle comeing from a Higgs bundle
$(E,\overline{\partial}_{E},\theta)$ via a harmonic metric $H$, then we call
$(E,D,H)$ is a harmonic bundle. Denote by $H_{DR}^{\ast}(X,E)$ and
$H_{Dol}^{\ast}(X,E)$ be the set of $L^{\infty}$-elements in the cohomology
groups of the complexes $(\Omega^{\ast}(E),D)$ and
$(\Omega^{\ast}(E),\overline{\partial}_{E,\theta})$ respectively.

######  Lemma 4.1.

Assume $(X,\omega)$ be a Kähler manifold satisfying Assumption 2.1.

  1. (1)

If $(E,D)$ be a flat bundle and $H$ is a harmonic metric, then any
$D$-parallel $L^{\infty}$-section is also
$D_{H}^{0,1}+\psi_{H}^{1,0}$-parallel.

  2. (2)

If $(E,\overline{\partial}_{E},\theta)$ be a Higgs bundle and $H$ is a
Hermitian-Einstein metric such that the
$\sqrt{-1}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,H}=0$, then any
$\overline{\partial}_{E,\theta}$-parallel $L^{\infty}$-section is also
$D_{\overline{\partial}_{E},\theta,H}$-parallel.

  3. (3)

If $(E,D,H)$ be a harmonic bundle, then $H_{DR}^{0}(X,E)=H_{Dol}^{0}(X,E)$.

###### Proof.

The statement $(3)$ follows from $(1)$ and $(2)$, Firstly suppose $f$ is
$D$-parallel, we have

(4.23)
$\begin{split}\Delta|f|_{H}^{2}&=\sqrt{-1}\Lambda_{\omega}(D_{H}^{1,0}D_{H}^{0,1}f,f)_{H}+\sqrt{-1}\Lambda_{\omega}(D_{H}^{1,0}f,D_{H}^{1,0}f)_{H}\\\
&-\sqrt{-1}\Lambda_{\omega}(D_{H}^{0,1}f,D_{H}^{0,1}f)_{H}+\sqrt{-1}\Lambda_{\omega}(f,D_{H}^{0,1}D_{H}^{1,0}f)_{H}\\\
&=|D_{H}^{0,1}f|_{H}^{2}+|\psi_{H}^{1,0}f|^{2}_{H}-\sqrt{-1}\Lambda_{\omega}(D_{H}^{1,0}\psi_{H}^{0,1}f,f)_{H}-\sqrt{-1}\Lambda_{\omega}(f,D_{H}^{0,1}\psi_{H}^{1,0}f)_{H}\\\
&=|D_{H}^{0,1}f|_{H}^{2}+|\psi_{H}^{1,0}f|^{2}_{H}+\sqrt{-1}\Lambda_{\omega}(\psi_{H}^{0,1}D_{H}^{1,0}f,f)_{H}+\sqrt{-1}\Lambda_{\omega}(f,\psi_{H}^{1,0}D_{H}^{0,1}f)_{H}\\\
&=|D_{H}^{0,1}f|_{H}^{2}+|\psi_{H}^{1,0}f|^{2}_{H}-\sqrt{-1}\Lambda_{\omega}(D_{H}^{1,0}f,\psi_{H}^{1,0}f)_{H}+\sqrt{-1}\Lambda_{\omega}(\psi_{H}^{0,1}f,D_{H}^{0,1}f)_{H}\\\
&=2|(D_{H}^{0,1}+\psi_{H}^{1,0})f|^{2}_{H}.\end{split}$

By Assumption 2.1 we have $f$ is $D_{H}^{0,1}+\psi_{H}^{1,0}$-parallel.

On the other hand, if $f$ is $\overline{\partial}_{E,\theta}$-parallel, we
deduce

(4.24)
$\begin{split}\Delta|f|_{H}^{2}&=\sqrt{-1}\Lambda_{\omega}(\partial_{H}f,\partial_{H}f)_{H}+\sqrt{-1}\Lambda_{\omega}(f,\overline{\partial}_{E}\partial_{H}f)_{H}\\\
&=|\partial_{H}f|_{H}^{2}+\sqrt{-1}\Lambda_{\omega}(f,F_{\overline{\partial}_{E},\theta,H}f)_{H}-\sqrt{-1}\Lambda_{\omega}(f,[\theta,\theta^{\ast
H}]f)_{H}\\\ &=|\partial_{\theta,H}f|_{H}^{2},\end{split}$

and then we see $f$ is $D_{\overline{\partial}_{E},\theta,H}$-parallel. ∎

Given a vector bundle $E$ over a complex manifold $(X,\omega)$, equipped with
a Hermitian metric $H$ and a connection $D$. The contraction of pseudo-
curvature operator can be written as

(4.25)
$\begin{split}\sqrt{-1}\Lambda_{\omega}G_{H}&=\frac{\sqrt{-1}}{4}\Lambda_{\omega}[D^{0,1}+\delta_{H}^{0,1},D^{1,0}-\delta_{H}^{1,0}]\\\
&=\frac{\sqrt{-1}}{4}\Lambda_{\omega}\left(D^{2}-(\delta_{H}^{1,0}+D^{0,1})^{2}+(D^{1,0}+\delta_{H}^{0,1})^{2}-\delta_{H}^{2}\right),\end{split}$

If $D$ is flat, we see

(4.26)
$\begin{split}\sqrt{-1}\Lambda_{\omega}G_{H}&=\frac{\sqrt{-1}}{4}\Lambda_{\omega}\left(-(\delta_{H}^{1,0}+D^{0,1})^{2}+(D^{1,0}+\delta_{H}^{0,1})^{2}\right)\\\
&=\frac{\sqrt{-1}}{4}\Lambda_{\omega}(-[D^{0,1},\delta_{K}^{1,0}+h^{-1}\delta_{K}^{1,0}h]+[\delta_{K}^{0,1}+h^{-1}\delta_{K}^{0,1}h,D^{1,0}])\\\
&=\sqrt{-1}\Lambda_{\omega}G_{K}+\frac{\sqrt{-1}}{4}\Lambda_{\omega}D(h^{-1}D_{K}^{c}h),\end{split}$

where $h=K^{-1}H$ and $D_{K}^{c}=\delta_{K}^{0,1}-\delta_{K}^{1,0}$.

Finally, we are in the position prove Theorem 1.5.

######  Proof of Theorem 1.5.

Given $[D]\in\mathcal{M}_{Flat,K,\phi}$, by Theorem 1.3, we yield a Poisson
metric $H$ with $\det h=1,|h|\in L^{\infty},||Dh||_{L^{2}}\leq
C||\phi||_{L^{1}}^{\frac{1}{2}}$ for a positive constant $C$ and $h=K^{-1}H$.
Then Proposition 4.3 implies
$(\overline{\partial}_{E},\theta)=(D_{H}^{0,1},\psi_{H}^{\perp,1,0})$
determines an irreducible Higgs structure with $c_{1}(E,H)=c_{1}(E,K)=0$.
Moreover, it can be seen $H$ is harmonic with respect to
$D_{H}+\psi_{H}^{\perp}$ and due to Lemma 4.1, the assignment
$[D]\mapsto[(\overline{\partial}_{E},\theta)]$ is well-defined. Then we
compute

(4.27)
$\begin{split}\sqrt{-1}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}&=\sqrt{-1}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,H}+\sqrt{-1}\Lambda_{\omega}\overline{\partial}_{E,\theta}(h\partial_{\theta,H}h^{-1})\\\
&=\sqrt{-1}\Lambda_{\omega}D_{\overline{\partial}_{E},\theta,H}(h\partial_{\theta,H}h^{-1})-\sqrt{-1}\Lambda_{\omega}\partial_{\theta,H}(h\partial_{\theta,H}h^{-1})\\\
&=-\sqrt{-1}\Lambda_{\omega}D_{\overline{\partial}_{E},\theta,H}(hD_{\overline{\partial}_{E},\theta,H}^{c}h^{-1})+\sqrt{-1}\Lambda_{\omega}D_{\overline{\partial}_{E},\theta,H}(h\overline{\partial}_{E,\theta}h^{-1})\\\
&-\sqrt{-1}\Lambda_{\omega}\partial_{\theta,H}(h\partial_{\theta,H}h^{-1})\\\
&=4\sqrt{-1}\Lambda_{\omega}G_{H}^{\perp}-4\sqrt{-1}\Lambda_{\omega}G_{K}^{\perp}+\sqrt{-1}\Lambda_{\omega}D_{\overline{\partial}_{E},\theta,H}(h\overline{\partial}_{E,\theta}h^{-1})\\\
&-\sqrt{-1}\Lambda_{\omega}\partial_{\theta,H}(h\partial_{\theta,H}h^{-1})\\\
&=-2(D_{K}^{\ast}\psi_{K})^{\perp}+\sqrt{-1}\Lambda_{\omega}\partial_{\theta,H}(h\overline{\partial}_{E,\theta}h^{-1})\\\
&+\sqrt{-1}\Lambda_{\omega}\overline{\partial}_{E,\theta}(h\overline{\partial}_{E,\theta}h^{-1})-\sqrt{-1}\Lambda_{\omega}\partial_{\theta,H}(h\partial_{\theta,H}h^{-1}),\end{split}$

and

(4.28)
$\begin{split}\sqrt{-1}\Lambda_{\omega}\partial_{\theta,H}(h\overline{\partial}_{E,\theta}h^{-1})&=\sqrt{-1}\Lambda_{\omega}(\partial_{\theta,H}h\wedge\overline{\partial}_{E,\theta}h^{-1})+\sqrt{-1}\Lambda_{\omega}(\overline{\partial}_{E,\theta}h\wedge\partial_{\theta,H}h^{-1})\\\
&-\sqrt{-1}\Lambda_{\omega}(\overline{\partial}_{E,\theta}h\wedge\partial_{\theta,H}h^{-1})-\sqrt{-1}\Lambda_{\omega}(h\overline{\partial}_{E,\theta}\partial_{\theta,H}h^{-1})\\\
&=\sqrt{-1}\Lambda_{\omega}(\partial_{\theta,H}h\wedge\overline{\partial}_{E,\theta}h^{-1})+\sqrt{-1}\Lambda_{\omega}(\overline{\partial}_{E,\theta}h\wedge\partial_{\theta,H}h^{-1})\\\
&-\sqrt{-1}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}.\end{split}$

where we have used $(\ref{sec428})$. We obtain from above two equalities that

(4.29)
$\begin{split}\sqrt{-1}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}&=-(D_{K}^{\ast}\psi_{K})^{\perp}+\frac{\sqrt{-1}}{2}\Lambda_{\omega}(\partial_{\theta,H}h\wedge\overline{\partial}_{E,\theta}h^{-1})\\\
&+\frac{\sqrt{-1}}{2}\Lambda_{\omega}(\overline{\partial}_{E,\theta}h\wedge\partial_{\theta,H}h^{-1})+\frac{\sqrt{-1}}{2}\overline{\partial}_{E,\theta}(h\overline{\partial}_{E,\theta}h^{-1})\\\
&-\frac{\sqrt{-1}}{2}\Lambda_{\omega}\partial_{\theta,H}(h\partial_{\theta,H}h^{-1}),\end{split}$

from which we conclude
$|\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}|\in L^{1}$.

For any $\theta$-invariant sub-holomorphic bundle $V\subset E$, we have the
orthogonal decomposition $E=V\oplus V^{\perp}$ with respect to $K$ and the
projection $\pi_{K}$ onto $V$. We denote by $K_{V}$, $H_{V}$ the metrics on
$V$ induced by $K$ and $H$. If $|\overline{\partial}_{E,\theta}\pi_{K}|\not\in
L^{2}$, by Chern-Weil formula $(\ref{ChernHiggs})$ and
$|\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}|\in L^{1}$, we know

(4.30) $\begin{split}\deg_{\omega}(V,K)=-\infty.\end{split}$

If $|\overline{\partial}_{E,\theta}\pi_{K}|\in L^{2}$, the same reason
indicates $\deg_{\omega}(V,K)$ is finite. It also holds

(4.31) $\begin{split}h_{V}=\pi_{K}\circ h\circ\pi_{K},\end{split}$ (4.32)
$\begin{split}\overline{\partial}_{V,\theta_{V}}h_{V}=\pi_{K}\circ\overline{\partial}_{E,\theta}h\circ\pi_{K}+\overline{\partial}_{E,\theta}\pi_{K}\circ(\operatorname{id}_{E}-\pi_{K})\circ
h\circ\pi_{K},\end{split}$

where $h_{V}=K_{V}^{-1}H_{V}$ and $\overline{\partial}_{V,\theta_{V}}$ is the
induced Higgs structure on $V$. Therefore

(4.33)
$\begin{split}||\operatorname{tr}(h_{V}^{-1}\partial_{\theta_{V},K_{V}}h_{V})||_{L^{2}}&\leq||h_{V}^{-1}||_{L^{\infty}}||\overline{\partial}_{V,\theta_{V}}h_{V}||_{L^{2}}<\infty.\end{split}$

So Simpson’s Lemma 5.2 in [39](or Yau’s Lemma in [46]) applies,

(4.34)
$\begin{split}\lim\limits_{j\rightarrow\infty}\int_{X_{j}}\sqrt{-1}\overline{\partial}\operatorname{tr}(h_{V}^{-1}\partial_{\theta_{V},K_{V}}h_{V})=0,\end{split}$

where $\\{X_{j}\\}$ is a sequence of exhaustion subsets. And

(4.35)
$\begin{split}\lim\limits_{j\rightarrow\infty}\int_{X_{j}}\sqrt{-1}\operatorname{tr}F_{\overline{\partial}_{V},\theta_{V},H_{V}}&=\lim\limits_{j\rightarrow\infty}\int_{X_{j}}\sqrt{-1}\operatorname{tr}F_{\overline{\partial}_{V},\theta_{V},K_{V}}\\\
&+\lim\limits_{j\rightarrow\infty}\int_{X_{j}}\sqrt{-1}\operatorname{tr}(\overline{\partial}_{V,\theta_{V}}(h_{V}^{-1}\partial_{\theta_{V},K}h_{V}))\\\
&=\deg_{\omega}(V,K),\end{split}$

then we have

(4.36)
$\begin{split}\lim\limits_{j\rightarrow\infty}\int_{X_{j}}\sqrt{-1}\operatorname{tr}F_{\overline{\partial}_{V},\theta_{V},H_{V}}&=\int_{X}\sqrt{-1}\operatorname{tr}F_{\overline{\partial}_{V},\theta_{V},H_{V}}=\deg_{\omega}(V,H).\end{split}$

Since $(\overline{\partial}_{E},\theta)$ is $H$-analytically stable, we
conclude by $(\ref{sec4210})$, $(\ref{sec4211})$ and $(\ref{sec4212})$ that
$(\overline{\partial}_{E},\theta)$ is $K$-analytically stable.

Next assume $\tilde{H}$ be another Hermitian-Einstein metric such that
$\det\tilde{h}=1$ and $|\tilde{h}|\in L^{\infty}$ for
$\tilde{h}=K^{-1}\tilde{H}$. Set $\hat{h}=H^{-1}\tilde{H}$, we have
$\sqrt{-1}\Lambda_{\omega}\partial\overline{\partial}\operatorname{tr}\hat{h}=|\hat{h}^{-\frac{1}{2}}\partial_{\theta,H}\hat{h}|_{H}^{2}$
and the Assumption 2.1 implies $\partial_{\theta,H}\hat{h}=0$. We let
$E=\mathop{\bigoplus}\limits_{j=1}^{m}E_{j}$ be the eigendecomposition of
$\hat{h}$, which is orthogonal with respect to $H$ and $\tilde{H}$. Moreover,
it holds $\overline{\partial}_{E,\theta}\pi_{j}=0$ for the projection
$\pi_{j}$ onto $E_{j}$. But $(\overline{\partial}_{E},\theta)$ is
$H$-analytically stable, we know $m=1$, $H=\tilde{H}$ and therefore
$[(\overline{\partial}_{E},\theta)]\in\mathcal{M}_{Higgs,K}$.

Conversely, given
$[(\overline{\partial}_{E},\theta)]\in\mathcal{M}_{Higgs,K,\phi}$, by
Mochizuki’s work [33] we know that there exists a Hermitian-Einstin metric $H$
with $\det h=1$, $|h|\in L^{\infty}$ and
$||\overline{\partial}_{E,\theta}h||_{L^{2}}\leq
C||\phi||_{L^{1}}^{\frac{1}{2}}$ for a positive constant $C$, where
$h=K^{-1}H$. Moreover, the Hitchin-Simpson connection
$D_{\overline{\partial}_{E},\theta,H}$ is flat and irreducible. Due to Lemma
4.1, the map
$[(\overline{\partial}_{E},\theta)]\mapsto[D_{\overline{\partial}_{E},\theta,H}]$
is well-defined. Next as $H$ is harmonic with respect to
$D=D_{\overline{\partial}_{E},\theta,H}$, a similar discussion as that in
$(\ref{sec429})$ shows

(4.37)
$\begin{split}D_{K}^{\ast}\psi_{K}&=-\sqrt{-1}\Lambda_{\omega}F_{\overline{\partial}_{E},\theta,K}+\frac{\sqrt{-1}}{2}\Lambda_{\omega}(\partial_{\theta,H}h\wedge\overline{\partial}_{E,\theta}h^{-1})\\\
&+\frac{\sqrt{-1}}{2}\Lambda_{\omega}(\overline{\partial}_{E,\theta}h\wedge\partial_{\theta,H}h^{-1})+\frac{\sqrt{-1}}{2}\overline{\partial}_{E,\theta}(h\overline{\partial}_{E,\theta}h^{-1})\\\
&-\frac{\sqrt{-1}}{2}\Lambda_{\omega}\partial_{\theta,H}(h\partial_{\theta,H}h^{-1}).\end{split}$

and therefore $|(D_{K}^{\ast}\psi_{K})^{\perp}|=|D_{K}^{\ast}\psi_{K}|\in
L^{1}$. Then proceeding the argument in the proof of the Theorem 1.3, we
conclude $D_{\overline{\partial}_{E},\theta,H}$ is analytically stable with
respect to the background metric $K$ and hence
$[D_{\overline{\partial}_{E},\theta,H}]\in\mathcal{M}_{Flat,K}$. ∎

###  4.3. Vanishing theorems of characteristic classes

Associated with a vector bundle $(E,D)$, up to some normalization
coefficients, the Kamber-Tondeur classes are defined by

(4.38) $\begin{split}\alpha_{2k+1}(E,D)=[\operatorname{tr}\psi_{H}^{2k+1}]\in
H^{2k+1}_{DR}(M,\mathbb{R}),\end{split}$

and the class is independent of the choice of fiber metrics, see [4, 11, 16,
23, 35] for the related study of these characteristic classes.

By Proposition 4.4, we know $\psi_{H}^{1,0}\wedge\psi_{H}^{1,0}=0$ and
following [24], for any $k\geq 1$ we deduce

(4.39)
$\begin{split}\operatorname{tr}\psi_{H}^{2k+1}&=\operatorname{tr}\left((\psi_{H}^{1,0}\wedge\psi_{H}^{0,1})^{2k}\wedge\psi_{H}^{1,0}\right)+\operatorname{tr}\left((\psi_{H}^{0,1}\wedge\psi_{H}^{1,0})^{2k}\wedge\psi_{H}^{0,1}\right)\\\
&=\operatorname{tr}\left(\psi_{H}^{1,0}\wedge(\psi_{H}^{1,0}\wedge\psi_{H}^{0,1})^{2k}\right)+\operatorname{tr}\left(\psi_{H}^{0,1}\wedge(\psi_{H}^{0,1}\wedge\psi_{H}^{1,0})^{2k}\right)\\\
&=0.\end{split}$

So we conclude the following vanishing theorem.

######  Theorem 4.1.

Let $(X,\omega)$ be a complete Kähler manifold with bounded Ricci curvature
from below and satisfying the Assumptions 2.1. Assume $(E,D)$ be a
$K$-analytically stable flat bundle over $X$ such that
$|(D_{K}^{\ast}\psi_{K})^{\perp}|\leq C\phi$ for a constant $C$ and
$|\psi_{K}|\in L^{2}$. Then the characteristic classes $\alpha_{2k+1}(E,D)$
vanish for all $k\geq 1$.

We mention that in the compact setting, this result was firstly proved in
[36].

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