Flat Higgs bundles over non-compact affine Gauduchon manifolds
Zhenghan Shen Chuanjing Zhang Xi Zhang







Zhenghan Shen
School of Mathematical Sciences

University of Science and Technology of China

Hefei, 230026,P.R. China
szh2015@mail.ustc.edu.cn

Chuanjing Zhang
School of Mathematical Sciences

University of Science and Technology of China

Hefei, 230026,P.R. China
chjzhang@mail.ustc.edu.cn
Xi Zhang
School of Mathematical Sciences

University of Science and Technology of China

Hefei, 230026,P.R. China
  mathzx@ustc.edu.cn

[]53C07, 57N16


Flat Higgs bundles  over non-compact affine Gauduchon manifolds
    Zhenghan Shen, Chuanjing Zhang and Xi Zhang
    December 30, 2023
===============================================================





In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.


0.2 true cm



myheadings

Z. Shen et al.
         Flat Higgs bundles  over non-compact affine Gauduchon manifolds











§  INTRODUCTION


An affine manifold is a smooth real manifold X equipped with a flat torsion-free connection D on its tangent bundle TX, or equivalently an affine structure which is provided by an atlas of X whose transition functions are affine maps of the form

    x↦ Ax+b,

where A∈ GL(n,ℝ) and b∈ℝ^n. An affine manifold X is called special if it admits a volume form ν which is covariant constant with respect to the flat connection D, or  equivalently, the holonomy group of D lies in the special affine subgroup, i.e. A ∈ SL(n,ℝ). Special affine manifolds play a role in the Strominger-Yau-Zaslow conjecture (<cit.>). Cheng and Yau (<cit.>) proved that on a closed special affine manifold, an affine Kähler metric (if it exists) can be deformed to a flat metric by adding the Hessian of a smooth function. This result shows that  the compact special affine manifolds are all torus quotients. A Riemannian metric g on an affine manifold is said to be affine Gauduchon if

    ∂∂̅(ω_g^n-1)=0,

where ∂,∂̅ and ω_g are defined in section <ref>. On a compact special affine manifold M, every conformal class of Riemannian metrics contains an affine Gauduchon metric, which is unique up to scaling by a constant (see <cit.>).
















A Higgs bundle (E,∂̅_E,θ) is a holomorphic vector bundle E coupled with a Higgs filed θ∈Ω_X^1,0( End(E)) such that ∂̅_Eθ=0 and θ∧θ =0. Higgs bundles were introduced by Hitchin (<cit.>) in his study of the self duality equations. They have rich structures and play an important role in many areas including gauge theory, Kähler and hyper-Kähler geometry, group representations and nonabelian Hodge theory. When (X,ω) is a compact Kähler manifold, Hitchin (<cit.>) and Simpson (<cit.>) obtained a Higgs bundle version of Donaldson-Uhlenbeck-Yau theorem (<cit.>, <cit.>, <cit.>, <cit.>), i.e. they proved that a Higgs bundle admits the Hermitian-Einstein metric if and only if it's Higgs poly-stable. Simpson (<cit.>) also considered some non-compact Kähler manifolds case. He introduced the concept of analytic stability for Higgs bundles and proved that the analytic stability implies the existence of Hermitian-Einstein metric. In <cit.>, the authors also proved a generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over a class of non-compact Gauduchon manifolds.

When the base space (X,D,g,ν) is a compact special affine manifold equipped with an affine Gauduchon metric g, Loftin (<cit.>) established a Donaldson-Uhlenbeck-Yau type theorem. He proved that if a flat complex vector bundle E over (X,D,g,ν) is stable, then there is an affine Hermitian-Einstein metric on E. Biswas, Loftin and Stemmler (<cit.>) also studied the flat Higgs bundle case. The definition of the flat Higgs bundle will be introduced in section <ref>. They proved  that a flat Higgs bundle over a compact special affine Gauduchon manifold  is poly-stable if and only if it admits an affine Hermitian-Einstein metric. There are many other interesting and important works related (<cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>, <cit.>).

In this paper, we study the non-compact and affine Gauduchon case. In the sequel, we always assume that (X,D,g,ν) is a special affine Gauduchon manifold unless otherwise stated. By Simpson(<cit.>), we will make the following three assumptions:

Assumption 1. (X,D,g,ν) has finite volume.

Assumption 2. There exists a non-negative exhaustion function ϕ with tr_g∂∂̅ϕ bounded.

Assumption 3. There is an increasing function a:[0,+∞)→ [0,+∞) with a(0)=0 and a(x)=x for x>1 such that if f is a bounded positive function on X with tr_g∂∂̅f≥ -B, then

    sup_X|f|≤ C(B)a(∫_X|f|ω^n_g/ν),

where B is a constant. Furthermore, if tr_g∂∂̅f≥ 0, then tr_g∂∂̅f=0.

In the non-compact and affine Gauduchon case, we fix a background metric H_0 on the flat Higgs bundle (E,∇,θ), where ∇ is the flat connection on E, and define the analytic degree of E to be a real number

    _g(E,H_0)=1/n∫_X tr( tr_g(F_H_0+[θ,θ^*H_0]))ω_g^n/ν,

where F_H_0 denotes the extended curvature form which is defined in section <ref>.
For the flat sub-Higgs bundle V⊂ E, we denote F_V by the extended curvature form of the extended Hermitian connection on V with respect to the Hermitian metric H_0|_V which is the restriction of H_0 to V. According to Chern-Weil formula, we can define the analytic degree of every θ-invariant flat sub-Higgs bundle V of (E,∇,θ) by

    _g(V,H_0)=1/n∫_X( tr(π tr_g(F _H_0+[θ,θ^*H_0]))-|D”π|_H_0^2)ω^n_g/ν,

where D”=∂̅+θ and π denotes the projection onto V with respect to the metric H_0. Following Simpson (<cit.>), we say that the flat Higgs bundle (E,∇,θ) is H_0-analytic stable (semi-stable) if for every proper θ-invariant flat sub-Higgs bundle V⊂ E, we have

    _g(V,H_0)/ rank(V)<(≤)_g(E,H_0)/ rank(E).



  An affine Hermitian-Einstein metric on a flat Higgs bundle (E,∇,θ) is a Hermitian metric H   satisfying the equation
  
    tr_g(F_H+[θ,θ^*H])=λ· Id_E,

  for some constant scalar λ, which is called the Einstein factor.


In this paper, we will show that, under some assumptions on the special affine Gauduchon manifold (X,D,g,ν), the stability of (E,∇,θ) implies the existence of affine Hermitian-Einstein metric on it, i.e.  we obtain the following Donaldson-Uhlenbeck-Yau type theorem.

Let (X,D,g,ν) be a non-compact special affine Gauduchon manifold satisfying the Assumptions 1,2,3 and 
|dω_g|_g∈ L^∞(X), (E,∇,θ) be a flat Higgs bundle over X with a background Hermitian metric H_0 satisfying sup_X| tr_g(F_H_0+[θ,θ^*H_0])|_H_0<+∞. If (E,∇,θ) is H_0-analytic stable, then there exists a metric H with D”(log H_0^-1H)∈ L^2, H and H_0  mutually bounded, such that

    tr_g(F_H+[θ,θ^*H])=λ· Id_E,

where the constant λ= deg_g(E,H_0)/ rank(E) Vol(X).


We should remark that Biswas, Loftin and Stemmler (<cit.>) only proved the Donaldson-Uhlenbeck-Yau type theorem in compact case. They used the continuity method and followed the argument of Lübke and Teleman (<cit.>). Here we first introduce the  affine Hermitian-Yang-Mills flow

    H^-1(t)∂ H(t)/∂ t=-4( tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E),

on Higgs bundles over affine Gauduchon manifolds and then we study the existence of short-time solutions and long-time solutions. In <cit.>,  Simpson used Donaldson's heat flow method to attack the existence problem of the Hermitian-Einstein metrics on Higgs bundles, and his proof relies on the properties of the Donaldson functional. However, the Donaldson functional is not well defined when g is only affine Gauduchon. So Simpson's argument is not applicable in our situation directly.  In this paper, we study the non-compact affine Gauduchon cases by using the heat flow method and avoid the Donaldson functional. The key is that we use the important identity (<ref>) instead of the Donaldson functional. For simplicity, we set

    Φ(H,θ)= tr_g(F_H+[θ,θ^*H])-λ·  Id_E.

Under the assumptions as that in Theorem <ref>, we can prove the following identity:

    ∫_X tr(Φ(H_0,θ)s)+⟨Ψ(s)(D”s),D”s⟩_H_0ω^n_g/ν=∫_X tr(Φ(H,θ)s)ω^n_g/ν,

where s=log(H_0^-1H) and

    Ψ(x,y)={   e^y-x-1/y-x,      x≠ y,
       1,      x=y.
    .

By the above identity (<ref>) and Loftin's result (<cit.>) that L_1^2 weakly flat sub-bundles are smooth flat sub-bundles, using the heat flow method, we can obtain the existence result of affine Hermitian-Einstein metric, see section 5 for details. In <cit.>, Jacob used the heat flow technique for Higgs bundle over compact Gauduchon manifold. It should be pointed out that our argument is new even in the compact Hermitian case.

This paper is organized as follows. In section 2, we give some introduction of the basic theory of (p,q)-forms with values in a flat vector bundle, affine Hermitian-Einstein metric and flat Higgs bundle. In section 3, we introduce the affine Hermitian-Yang-Mills flow on Higgs bundles over affine Gauduchon manifolds and give some basic estimates of the flow which will be used in section 4. Then we study the existence of short-time solutions and long-time solutions when the base space  is compact. And we also solve the Dirichlet problem for the Hermitian-Einstein equation on a flat Higgs bundle. In section 4, we get the existence of the long-time solutions of the affine Hermitian-Yang-Mills flow when the base manifold is noncompact. In section 5, we  consider the stable case and complete the proof of Theorem <ref>.



Acknowledgement: The authors are partially supported by NSF in China No.11625106, 11571332 and 11721101. The second 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.





§ PRELIMINARY



 §.§ Affine Dolbeault complex

Let (M,D) be an affine manifold of dimension n, meaning that D is a flat torsion-free connection on the tangent bundle TM. The affine structure of M determines a natural complex structure of TM. In local affine coordinates {x^1,⋯,x^n}, the fact that every tangent vector Y∈ TM can be written as Y=y^i∂/∂ x^i induces the local coordinates {y^1,⋯ y^n} on TM. Then z^i=x^i+√(-1)y^i form the complex coordinates on TM, which  is usually denoted by M^ℂ. On M, there are natural (p,q)-forms (see <cit.>), which are the restriction of (p,q)-forms from M^ℂ. Let T^*M be the cotangent bundle over M. We denote by (
Λ^pT^*M)⊗ (Λ^qT^*M) the tensor product of vector bundles Λ^p(T^*M) and Λ^q(T^*M), and by 𝒜^p,q the space of all smooth sections of Λ^pT^*M⊗Λ^qT^*M.

If {x^i}_i=1^n are the local affine coordinates on M with respect to the connection D, then we will denote the induced frame on 𝒜^p,q as

    {dz^i_1∧⋯∧ dz^i_p⊗ dz̅^j_1∧⋯∧ dz̅^j_q},

where z^i=x^i+√(-1)y^i are the complex coordinates on M^ℂ. Note that dz^i=dz̅^i=dx^i on M.

There is a natural restriction map from Λ^p,q(M^ℂ) to 𝒜^p,q as follows:
for ψ∈Λ^p,q(M^ℂ), in local affine coordinates on an open subset U⊂ M, we have

    ∑ψ_i_1⋯ i_p,j_1⋯ j_q(dz^i_1∧⋯∧  dz^i_p)∧(dz̅^j_1∧⋯∧ dz̅^j_q)
       ↦∑ψ_i_1⋯ i_p,j_1⋯ j_q|_U(dz^i_1∧⋯∧ dz^i_p)⊗(dz̅^j_1∧⋯∧ dz̅^j_q),

where ψ_i_1⋯ i_p,j_1⋯ j_q are smooth functions on TU⊂ TM=M^ℂ, U is considered as the zero section of π:TU → U, that is for a function f on M^ℂ, f(x,y)|_U=f(x,0). And the sums are taken over all 1≤ i_1<⋯<i_p≤ n and 1≤ j_1<⋯<j_q≤ n.

We can also define the natural operators ∂ and ∂̅ acting on (p,q)-forms on M

    ∂ :𝒜^p,q→𝒜^p+1,q  and  ∂̅:𝒜^p,q→𝒜^p,q+1

by

    ∂ :=1/2(d^D⊗ Id)  and  ∂̅:=(-1)^p1/2( Id⊗ d^D),

where d^D is the exterior derivative for the forms of M induced from D and Id is the identity operator. In local affine coordinates, for ϕ=ϕ_i_1⋯ i_p,j_1⋯ j_q(dz^i_1∧⋯∧ dz^i_p)⊗ (dz̅^j_1∧⋯∧ dz̅^j_q)∈𝒜^p,q,

    ∂ϕ:=1/2(dϕ_i_1⋯ i_p,j_1⋯ j_q)∧ (dz^i_1∧⋯∧ dz^i_p)⊗ (dz̅^j_1∧⋯∧ dz̅^j_q)

and

    ∂̅ϕ:=(-1)^p1/2(dz^i_1∧⋯∧ dz^i_p)⊗ (dϕ_i_1⋯ i_p,j_1⋯ j_q)∧(dz̅^j_1∧⋯∧ dz̅^j_q).

These operators are the restrictions of the corresponding ∂ and ∂̅ operators on M^ℂ with respect to the restriction map above.

There is also a natural wedge product on 𝒜^p,q, which we take to be the restriction of the wedge product on M^ℂ: if ϕ_i⊗ψ_i ∈𝒜^p_i,q_i for i=1,2, then we define

    (ϕ_1⊗ψ_1)∧(ϕ_2⊗ψ_2):=(-1)^q_1p_2(ϕ_1∧ϕ_2)⊗ (ψ_1∧ψ_2)∈𝒜^p_1+p_2,q_1+q_2.


We can also define the conjugation operator on (p,q)-forms on M, which is the restriction of the complex conjugation on M^ℂ: if ϕ∈Λ^p(T^*M) and ψ∈Λ^q(T^*M) are complex valued forms, then we define

    ϕ⊗ψ:=(-1)^pqψ̅⊗ϕ̅.


Given a smooth Riemannian metric g on an affine manifold M, there is a natural nondegenerate (1,1)-form ω_g on M expressed in local affine coordinate as

    ω_g=∑_i,j=1^ng_ijdz^i⊗ dz̅^j.

The Riemannian metric g naturally is extended to a Hermitian metric on M^ℂ and ω_g is the restriction of the corresponding (1,1)-form on M^ℂ.








In the case of special affine structures, ν induces the natural maps

    𝒜^n,q→Λ^q(T^*M),  ν⊗χ↦ (-1)^n(n-1)/2χ

and

    𝒜^p,n→Λ^p(T^*M),  χ⊗ν↦ (-1)^n(n-1)/2χ,

which are called division by ν. In particular, any (n,n)-form χ can be integrated by considering the integral of χ/ν(see <cit.>).

Let (E,∇) be a flat complex vector bundle of rank r over an affine manifold M. As we can see, the pull back of E to TM=M^ℂ by the natural projection π:TM→ M will be denoted by E^ℂ. The flat connection ∇ is pulled back to a flat connection π^*∇ on E^ℂ. This flat vector bundle (E^ℂ,π^*∇) over M^ℂ  can be considered as extension of the flat vector bundle (E,∇) on the zero section of TM.

Let {s_1,⋯, s_r} be the locally constant frames on E with respect to the flat connection ∇, meaning that ∇(s_α)=0, for α=1,2,⋯ ,r. Any locally constant frame {s_1,⋯,s_r} of E over M can be extended to a locally constant frame of E^ℂ over M^ℂ, and they are also holomorphic frames on E^ℂ→ M^ℂ with respect to the holomorphic structure of E^ℂ induced by ∇.

Let H be a Hermitian metric on E. It defines a Hermitian metric on E^ℂ and let D_H be the Chern connection associated to this Hermitian metric on E^ℂ. Then according to the decomposition of (1,0)-part and (0,1)-part, D_H corresponds to a pair

    (∂_H,∂̅)=(∂_H,∇,∂̅_∇),

where

    ∂_H,∇:Γ(E)→𝒜^1,0(E)  and  ∂̅_∇: Γ(E)→𝒜^0,1(E)

are smooth differential operators defined as that in <cit.>. The pair (∂_H,∂̅) is called the extended Hermitian connection of (E,H).

For locally constant frame {s_1,⋯,s_r}, if we write H_αβ̅=H(s_α,s_β), then we can locally define the extended connection form

    A_H=H^-1∂ H∈𝒜^1,0( End E),

the extended curvature form

    F_H=∂̅A_H=∂̅(H^-1∂ H)∈𝒜^1,1( End E),

the extended mean curvature

    K_H= tr_gF_H∈ C^∞(M, End E),

and the extended first Chern form

    c_1(E,H)= tr F_H ∈𝒜^1,1,

which are the restrictions of the corresponding objects on E^ℂ. Here tr_g denotes the contraction of differential forms using Riemannian metric g, and tr denotes the trace map on the fibers of End E. Note that the extended first Chern form is given locally by

    c_1(E,H)=-∂∂̅(log(H_αβ̅)).










The degree of (E,∇) with respect to an affine Gauduchon metric g on M is defined to be

    deg_g(E):=∫_Mc_1(E,H)∧ω_g^n-1/ν,

where H is any Hermitian metric on E. Since c_1(E,H)=c_1(E,K)+∂∂̅(log(K_αβ̅)/(H_αβ̅)), this is well defined by <cit.> when M is compact.













1 true cm



 §.§ Flat Higgs vector bundles



  Let (E,∇) be a smooth vector bundle E equipped with a flat connection ∇ over an affine manifold M. A flat Higgs field θ on (E,∇) is defined to be a smooth section of T^*M⊗ End(E) such that
  
    
  (1) θ is covariant constant, that is d^∇(θ)=0, where
    
    d^∇:Γ(T^*M⊗ End E)⟶Γ((∧ ^2T^*M)⊗ End E)

    is the induced covariant derivation.
    
  (2)θ∧θ=0.
  
  If θ is a flat Higgs field on (E,∇), then (E,∇,θ) is called a flat Higgs bundle.


A Higgs field θ will always be understood as an element of 𝒜^1,0( EndE), meaning it is expressed in local affine coordinates as

    θ=∑_i=1^nθ_i⊗ dz^i,

where θ_i are locally defined flat sections of End(E) and note that dz^i=dx^i on M.

Given a Hermitian metric H  on (E,θ), the adjoint θ^*H of θ with respect to H will be regarded as an element of 𝒜^0,1( End E). In locally affine coordinates, this means that

    θ^*H=∑_j=1^n(θ_j)^*H⊗ dz̅^j.


In particular, the Lie bracket [θ,θ^*H] ∈𝒜^1,1( EndE) is locally written as

    [θ,θ^*H]=θ∧θ^*H+θ^*H∧θ=∑_i,j=1^n(θ_i∘(θ_j)^*H-(θ_j)^*H∘θ_i)⊗ dz^i⊗ dz̅^j.


The extended connection form A_H,θ of the  flat Higgs bundle (E,∇,θ,H) is defined by

    A_H,θ:=(A_H+θ,θ^*H)∈𝒜^1,0( EndE)⊕𝒜^0,1( EndE).

This extended connection form corresponds to the connection form of the Hitchin-Simpson connection D_H,θ=D_H+θ+θ^*H on E^ℂ→ M^ℂ. Analogously, the extended curvature form F_H,θ of (E,∇,θ,H) is defined to be

    F_H,θ   :=(∂_Hθ,∂̅A_H+[θ,θ^*H],∂̅(θ^*H))
       ∈𝒜^2,0( EndE)⊕𝒜^1,1( End E)⊕𝒜^0,2( EndE).

It corresponds to the curvature form F_H,θ=F_H+[θ,θ^*H]+∂_Hθ+∂̅θ^*H of the connection D_H,θ on E^ℂ. As in usual case, the extended mean curvature K_H,θ of (E,∇,θ,H) is obtained by contracting the (1,1)-part of the extended curvature F_H,θ using the Riemannian metric g, i.e.

    K_H,θ:= tr_g(∂̅A_H+[θ,θ^*H])∈𝒜^0,0( EndE).

Since tr[θ,θ^*H]=0, we have trK_H,θ= trK_H. So the extended mean curvature K_H,θ of (E,∇,θ,H) is also related to the first Chern form c_1(E,H) by

    ( trK_H,θ)ω_g^n=nc_1(E,H)∧ω_g^n-1.


























§ THE AFFINE HERMITIAN-YANG-MILLS FLOW ON HIGGS BUNDLES OVER COMPACT AFFINE GAUDUCHON MANIFOLDS




 §.§ The affine Hermitian-Yang-Mills flow

Let (M,D,g) be an n-dimensional compact affine  manifold equipped with an affine Gauduchon metric g, (E,∇,θ) be a rank r flat Higgs bundle over M with a background metric H_0. Consider the evolution equation

    H^-1(t)∂ H(t)/∂ t=-4( tr_g (F_H(t)+[θ,θ^*H(t)])-λ· Id_E).

We call it the affine Hermitian-Yang-Mills flow. Choosing a local affine coordinates system {x^i}_i=1^n on M, we define the affine Laplace operator for a function f as

    Δ̃f=4 tr_g∂∂̅f=g^ij∂ ^2f/∂ x^i∂ x^j,

where (g^ij) is the inverse matrix of the matrix (g_ij). As usual, we denote the Beltrami Laplacian operator by

    Δ f=g^ij∂ ^2 f/∂ x^i∂ x^j+[∂ g^ij/∂ x^i+g^kjΓ^i_ik]∂ f/∂ x^j.

The difference between the two Laplacians is given by a first order differential operator as follows

    (Δ̃-Δ)f=⟨ V,∇ f⟩_g,

where V is a well-defined vector field on M, which is locally expressed as V=V_l∂/∂ x^l=(∂ g^il/∂ x^i+g^klΓ^i_ik)∂/∂ x^l.

Take a locally constant frame s_α(1≤α≤ r) on E, and express H(t), F_H(t), θ, θ^*H(t) locally. When there is no confusion, we will omit parameter t and simply write H, F_H, h for H(t), F_H(t), h(t), respectively. Then the affine Hermitian-Yang-Mills flow (<ref>) can be rewritten as follows:

    ∂ H(t)/∂ t   =-4 tr_g∂̅∂ H+4 tr_g∂̅H H^-1∂ H +4(λ H- tr_g[θ,H^-1θ̅^TH])
       =Δ̃H+4 tr_g∂̅HH^-1∂ H+4(λ H- tr_g[θ,H^-1θ̅^TH]).

From (<ref>), one can see that the flow is a non-linear strictly parabolic equation.


  Let H(t) be a solution of the affine Hermitian-Yang-Mills flow (<ref>), then
  
    (∂/∂ t-Δ̃)( tr( tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E))=0

  and
  
    (∂/∂ t-Δ̃)| tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E|_H(t)^2≤ 0.



  Set h(t)=H_0^-1H(t). For simplicity, we denote Φ(t)=Φ(H(t),θ)= tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E and omit t in the calculations.

  Using the identities
  
    ∂_H-∂_H_0=h^-1∂_H_0h,

   
    F_H-F_H_0=∂̅_E(h^-1∂_H_0h),

    
    θ^*H=h^-1θ^*H_0h,

  we have
  
    ∂/∂ tΦ(H,θ)   =∂/∂ t( tr_g(F_H+[θ,θ^*H])-λ· Id_E)
       = tr_g(∂̅_E(∂_H(h^-1∂ h/∂ t))+[θ,[θ^*H,h^-1∂ h/∂ t]])

  and
  
    Δ̃|Φ(H,θ)|_H^2=   -4_g∂̅∂ tr(Φ H^-1Φ̅^TH)
    
        =   -4 tr_g∂̅ tr(∂Φ H^-1Φ̅^TH-Φ H^-1∂ H H^-1Φ̅^TH
       +Φ H^-1∂̅Φ^TH+Φ H^-1Φ̅^TH H^-1∂ H)
    
        =   2 Re⟨-4 tr_g∂̅_E∂_HΦ,Φ⟩_H+4|∂_HΦ|_H^2+4|∂̅_EΦ|_H^2.

According to (<ref>), one can easily check that

    ∂/∂ t trΦ= tr(∂Φ/∂ t)= tr_g( tr∂̅∂ (h^-1∂ h/∂ t) )=Δ̃ trΦ.

This implies (<ref>).

A simple computation gives us that

    ∂/∂ t|Φ(H,θ)|_H^2   =∂/∂ t tr(Φ H^-1Φ̅^TH)
       =2 Re⟨-4 tr_g∂̅_E∂_HΦ,Φ⟩_H-8⟨ tr_g[θ,[θ^*H,Φ]],Φ⟩_H.

From (<ref>) and (<ref>), we can conclude that

    (∂/∂ t-Δ̃)|Φ(H,θ)|_H^2   =-8⟨ tr_g[θ,[θ^*H,Φ]],Φ⟩_H-4|∂_HΦ|_H^2-4|∂̅_EΦ|_H^2
       ≤ 0.





Next, we will recall the Donaldson's distance on the space of the Hermitian metrics.

  For any two Hermitian metrics H and K on the complex vector bundle E, we define
  
    σ(H,K)= tr(H^-1K)+ tr(K^-1H)-2 rank E.



It is obvious that σ(H,K)≥ 0 with the equality if and only if H=K. A sequence of Hermitian metrics H_i converge to H in the C^0-topology if and only if sup_Mσ(H_i,H)→ 0 as i→∞.

Denote h=K^-1H, then we know

    tr(h( tr_g F_H- tr_g F_K))=-1/4Δ̃ tr h+ tr(- tr_g∂̅h h^-1∂_Kh).


Similarly,

    tr(h^-1( tr_gF_K- tr_gF_H))=-1/4Δ̃ tr h^-1+ tr(- tr_g∂̅h^-1h∂_Hh^-1).



So we also have the following proposition.


  Let H and K be two affine Hermitian-Einstein metrics, then σ(H,K) is subharmonic with respect to the affine Laplace operator, i.e.
  
    Δ̃σ(H,K)≥ 0.



Let H(t) and K(t) be two solutions of the affine Hermitian-Yang-Mills flow (<ref>), and denote h(t)=K^-1(t)H(t), then using  (<ref>) and (<ref>), we obtain

    Δ̃ tr h(t)=4( tr(- tr_g∂̅h h^-1∂_Kh)- tr(h( tr_gF_H- tr_gF_K)))

and

    Δ̃ tr h^-1(t)=4( tr(- tr_g∂̅h^-1h∂_Hh^-1)- tr(h^-1( tr_gF_K- tr_gF_H))).

By direct calculations, one can get

    ∂/∂ t tr h(t)   = tr(∂ h/∂ t)= tr(∂/∂ t(K^-1H))
       = tr(-K^-1∂ K/∂ tK^-1H+K^-1∂ H/∂ t)
       =-4 tr(h( tr_gF_H- tr_gF_K))-4 tr(h( tr_g[θ,θ^*H-θ^*K])).

Similarly,

    ∂/∂ t tr h^-1=-4 tr(h^-1( tr_gF_K- tr_gF_H))-4 tr(h^-1( tr_g[θ,θ^*K-θ^*H])).

From (<ref>)-(<ref>), it follows that

    (Δ̃-∂/∂ t)( tr h(t)+ tr h^-1(t))
    
      =   4 tr(- tr_g∂̅hh^-1∂_Kh)+4 tr(- tr_g∂̅h^-1h∂_Hh^-1)
       +4 tr(h( tr_g[θ,θ^*H-θ^*K]))+4 tr(h^-1( tr_g[θ,θ^*K-θ^*H]))
    ≥   0,

where we have used

    tr(h( tr_g[θ,θ^*H-θ^*K]))=|θ h^1/2-hθ h^-1/2|_K^2

and

    tr(h^-1( tr_g[θ,θ^*K-θ^*H]))=|h^-1/2θ-h^1/2θ h^-1|_K^2.

So we have proved the following proposition.

  Let H(t) and K(t) be two solutions of affine Hermitian-Yang-Mills flow (<ref>), then
  
    (Δ̃-∂/∂ t)σ(H(t),K(t))≥ 0.






 §.§ The long-time existence of the affine Hermitian-Yang-Mills flow 


Let (M,D,g) be a compact affine Gauduchon manifold (with possibly non-empty boundary), and (E,∇,θ) be a flat Higgs bundle with the initial Hermitian metric H_0. If M is closed, then we consider the following evolution equation:

    {   H^-1(t)∂ H(t)/∂ t=-4( tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E),
       H(0)=H_0.
      .

If M is a compact manifold with non-empty smooth boundary ∂ M, and the affine Gauduchon metric g is smooth and non-degenerate on the boundary, for given data H̃ on ∂ M, we consider the following Dirichlet boundary value problem:

    {   H(t)^-1∂ H(t)/∂ t=-4( tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E),
       H(0)=H_0,
       H|_∂ M=H̃,
      .

where H_0 satisfies the boundary condition. Based on the formula (<ref>), we know the above equations are non-linear strictly parabolic equation, so the standard parabolic theory gives the short-time existence:


  For sufficiently small T>0, the equations (<ref>) and (<ref>) have a smooth solution defined for 0<t<T.



Next, according to the arguments of <cit.>, <cit.> and <cit.>, we can show the long-time existence.

 (<cit.>)
  Suppose that a smooth solution H(t) of the evolution equation (<ref>) or (<ref>) is defined for 0≤ t <T<+∞. Then H(t) converge in C^0-topology to some continuous non-degenerate metric as t→ T.



  Given ϵ >0, by the continuity at t=0 we can find a δ such that
  
    sup_Mσ(H(t_0),H(t_0'))<ϵ

  for 0<t_0,t_0'<δ. Then Proposition <ref> and the maximum principle imply that
  
    sup_Mσ(H(t),H(t'))<ϵ

  for all t,t'> T-δ. This means that H(t) are uniform Cauchy sequence and converge to a continuous limiting metric H_T. On the other hand, by Proposition <ref>, we know that
  
    sup_M×[0,T)| tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E|<C,

  where C is a uniform constant only depending on the initial data H_0. A direct calculation yields
  
    |∂/∂ t(log tr h(t))|≤ 4| tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E|_H(t)

  and
  
    |∂/∂ t(log tr h^-1(t))|≤ 4| tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E|_H(t).

  Using above formulas, one can conclude that σ(H(t),H_0) are bounded uniformly on M×[0,T), therefore H_T is a non-degenerate metric.


Then we can also obtain the following results in the affine Gauduchon case.


   (<cit.>)Let (M,D,g) be a compact affine Gauduchon manifold without boundary (with non-empty boundary), (E,∇,θ) be the flat Higgs bundle with the initial Hermitian metric H_0. Let H(t), 0≤ t<T, be any one-parameter family of Hermitian metrics on (E,∇,θ) (and satisfy the Dirichlet boundary condition). If H(t) converge in C^0-topology to some continuous metric H_T as t→ T, and if sup_M| tr_gF_H(t),θ|_H_0 is bounded uniformly in t, then H(t) are bounded in C^1 and also bounded in L_2^p ( for any 1<p<+∞)  uniformly in t.


(<cit.>)
  The evolution equations (<ref>) and (<ref>) have a unique solution H(t) which exists for 0≤ t<+∞.









 §.§ The Dirichlet boundary problem for affine Hermitian-Einstein metric

Since we have proved the long-time existence of (<ref>), it remains for us to show that H(t) will converge to the affine Hermitian-Einstein metric which we want. In this section, we will discuss the Dirichlet boundary problem for affine Hermitian-Einstein metric by using the heat equation method to deform an arbitrary initial metric to the desired one. Similar to the discussions of Donaldson <cit.> and Simpson <cit.>, we also proved the following theorem.

  Let (E,∇, θ) be a flat Higgs bundle over the compact affine manifold M with non-empty boundary ∂ M. Then for any Hermitian metric H̃ on the restriction of E to ∂ M, there
is a unique  affine Hermitian-Einstein metric H on E such that H=H̃ on ∂ M.


Suppose H(t) is the solution of (<ref>) for 0≤ t<∞. By direct calculations, one can easily check that

    |d|η|_H|^2≤ |∇_Hη|_H^2

for any section η∈Γ( End(E)).
This together with the formula (<ref>) in Proposition <ref> gives out

    (Δ̃-∂/∂ t)| tr_g(F_H(t)+[θ,θ^*H(t)])-λ Id_E|_H(t)≥ 0.

According to <cit.>, we can solve the following Dirichlet problem on M:

    {   Δ̃v=-| tr_g(F_H_0+[θ,θ^*H_0])-λ Id_E|_H_0,
       v|_∂ M=0.
      .

Then set w(x,t)=∫_0^t| tr_g(F_H+[θ,θ^*H])-λ Id_E|_H(x,s)ds-v(x), where v is the solution of the Dirichlet problem (<ref>). From (<ref>), (<ref>) and the boundary condition satisfied by H(t), we see that | tr_g(F_H+[θ,θ^*])-λ Id_E|_H(x,t) vanishes on ∂ M for t>0. Then clearly w(x,t) satisfies

    {   (Δ̃-∂/∂ t)w(x,t)≥ 0,
       w(x,0)=-v(x),
       w(x,t)|_∂ M=0.
      .

Applying the maximum principle, we have

    ∫_0^t| tr_g(F_H+[θ,θ^*H])-λ Id_E|_H(x,s)ds ≤sup_y∈M v(y),

for any x ∈M and 0< t ≤∞.

Let 0≤ t_1≤ t, ĥ(t)=H^-1(t_1)H(t). It is obvious that

    ĥ^-1(t)∂ĥ(t)/∂ t=-4( tr_g(F_H(t)+[θ,θ^*H(t)])-λ Id_E).

Then

    ∂/∂ tlog tr(ĥ(t)) ≤ 4| tr_g(F_H(t)+[θ,θ^*H(t)])-λ Id_E|_H(t).

Integrating this from t_1 to t gives us

    tr(H^-1(t_1)H(t))≤ rexp(4∫_t_1^t| tr_g(F_H+[θ,θ^*H])-λ Id_E|_H(s)ds).

Similarly, we can get the estimate for tr(H^-1(t)H(t_1)). Combining them together, we can deduce that

    σ(H(t),H(t_1))≤ 2r(exp(4∫_t_1^t| tr_g(F_H+[θ,θ^*H])-λ Id_E|_H(s)ds)-1).

 Because of (<ref>), we conclude that H(t)  converge in C^0-topology to some continuous metric H_∞ as t→∞. Using Lemma <ref>, we know that H(t) have uniform C^1 and L_2^p(1<p<∞) bounds. On the other hand, |H^-1(t)∂ H(t)/∂ t| is bounded uniformly. This together with the standard elliptic regularity shows that there exists a subsequence H(t)→ H_∞ in C^∞-topology as t→∞. Of course (<ref>) means that

    tr_g(F_H_∞+[θ,θ^*H∞])=λ Id_E,

i.e. H_∞ is the desired affine Hermitian-Einstein metric satisfying the boundary condition. From Proposition <ref> and the maximum principle, the uniqueness of the solution follows.







§ THE AFFINE HERMITIAN-YANG-MILLS FLOW ON NON-COMPACT AFFINE GAUDUCHON MANIFOLD


In this section, we will discuss the affine Hermitian-Yang-Mills flow on some non-compact manifold. Before the discussion, we need some important propositions. First, we derive the local C^1-estimate. Then, we want to get following identity in the non-compact affine Gauduchon case:

    ∫_M tr(Φ(H_0,θ)s)ω^n_g/ν+∫_M⟨Ψ(s)(D”s),D”s⟩_H_0ω^n_g/ν=∫_M tr(Φ(H,θ)s)ω^n_g/ν.



Consider the following affine Hermitian-Yang-Mills flow

    H^-1(t)∂ H(t)/∂ t=-4( tr_g(F_H(t)+[θ,θ^*H(t)])-λ· Id_E).



  Suppose H(t) is a long-time solution of the flow (<ref>) on the compact affine manifold M(with nonempty smooth boundary ∂ M). Set h(t)=H_0^-1H(t) and assume there exists a constant C̅_0 such that
  
    sup_(x,t)∈M×[0,+∞)|log h(t)|_H_0≤C̅_0.

  Then, for any compact subset Ω⊂M, there exists a uniform constant C̅_1 depending only on C̅_0,d^-1 and  the geometry of Ω̃ such that
  
    sup_(x,t)∈Ω×[0,+∞)|h^-1(t)∂_H_0h(t)|_H_0≤C̅_1,

  where d is the distance of Ω to ∂ M, and Ω̃={x∈M| dist(x,Ω)≤1/2d}.



  We will follow the argument in <cit.> to get the local uniform C^1-estimate. For simplicity, we denote 𝒯(t)=h^-1(t)∂_H_0h(t) and Φ(H(t),θ)= tr_gF_H(t),θ-λ· Id_E. Then the direct computations give us that
  
    (Δ̃-∂/∂ t) trh   =4 tr(- tr_g∂̅hh^-1∂_H_0h)+4 tr(hΦ(H_0,θ))
        +4 tr(h[θ,θ^H-θ^*H_0])
       =4 tr(- tr_g∂̅hh^-1∂_H_0h)+4 tr(hΦ(H_0,θ))
        +4 tr([θ,h]∧ h^-1[h,θ^*H_0])
       ≥ 4 tr(- tr_g∂̅hh^-1∂_H_0h)+4 tr(hΦ(H_0,θ)),

  
    ∂/∂ t𝒯=∂/∂ t(H^-1∂ H)=∂_H_0(h^-1∂ h/∂ t)=-4∂_H(Φ(H,θ)),

 and

    (Δ̃-∂/∂ t)|𝒯|_H^2≥    |∇_H𝒯|_H^2-Ĉ_1(| tr_gF_H_0|_H+|F_H_0|_H+|θ|_H^2+|Ric(g)|_g+|∇_gJ|)|𝒯|^2_H
       -Ĉ_2|∇_H_0 tr_gF_H_0|_H|𝒯|_H-4|∇_H_0θ|_H^2,

  where J is the complex structure on M^ℂ and  positive constants Ĉ_1,Ĉ_2 depend only on the dimension n and rank r. By (<ref>) and Proposition <ref>, we have
  
    (Δ̃-∂/∂ t)|𝒯|_H^2≥ |∇_H𝒯|_H^2-Ĉ_3|𝒯|_H^2-Ĉ_3

  on the domain Ω̃×[0,+∞), where Ĉ_3 is a uniform constant depending only on C̅_0, sup_Ω̃| tr_gF_H_0|_H_0, sup_Ω̃|F_H_0|_H_0, sup_Ω̃|∇_H_0 tr_gF_H_0|_H_0, sup_Ω̃|∇_H_0θ|_H_0, sup_Ω̃|θ|_H_0 and the geometry of Ω̃.

 Set Ω̅={x∈M|  dist(x,Ω) ≤1/4d}. Let φ_1, φ_2 be  the non-negative cut-off functions satisfying:
  
    φ_1={
        1,     x∈Ω,
    
        0,     x∈ M∖Ω̅,
      .

  
    φ_2={
        1,     x∈Ω̅,
    
        0,     x∈ M∖Ω̃,
      .

 and
 
    |dφ_i|^2+|Δ̃φ_i|≤ c,i=1,2,

where c is a constant depending only on d^-2. Consider the following test function

    f(·,t)=φ_1^2|𝒯|_H^2+Wφ_2^2 trh,

where the constant W will be chosen large enough later. It follows from (<ref>) and (<ref>) that

    (Δ̃-∂/∂ t)f≥   φ_1^2(|∇_H𝒯|_H^2-Ĉ_3|𝒯|_H^2-Ĉ_3)+Δ̃φ_1^2|𝒯|_H^2
       +4⟨φ_1∇φ_1,∇|𝒯|_H^2⟩
       +WΔ̃φ_2^2 trh+4W⟨φ_2∇φ_2,∇ trh⟩
       +4Wφ_2^2 tr( tr_gh^-1∂_H_0h∂̅h+hΦ(H_0,θ)).

Noting that

    4⟨φ_1∇φ_1,∇|𝒯|_H^2⟩   ≥ -8φ_1|∇φ_1||𝒯|_H|∇_H𝒯|_H
       ≥-φ_1^2|∇_H𝒯|_H^2-16|∇φ_1|^2|𝒯|_H^2,


    2W⟨φ_2∇φ_2,∇ trh⟩≥-φ_2^2|∇ trh|^2_H-W^2|∇φ_2|^2,

and

    |𝒯|^2_H   = tr( tr_gh^-1∂_H_0hH^-1(h^-1∂_H_0h)^TH)
       = tr( tr_gh^-1∂_H_0h h^-1∂̅h)
       ≤ e^C̅_0 tr( tr_gh^-1∂_H_0h∂̅h).

Then there holds that

    (Δ̃-∂/∂ t)f≥φ_2^2(-Ĉ_3-18c-4e^2C̅_0+4We^-C̅_0)|𝒯|_H^2-C̃_0,

where C̃_0 is a positive constant depending only on C̅_0 and Ĉ_3. If we choose

    W=1/4e^-C̅_0(Ĉ_3+18c+4e^2C̅_0+1),

then we can obtain

    (Δ̃-∂/∂ t)f≥φ_2^2|𝒯|^2_H-C̃_0

on M×[0,+∞). Let f(q,t_0)=max_M×[0,+∞)f. On the basis of the definition of φ_i and the local uniform C^0-bound of h(t), we may assume that

    (q,t_0)∈M×(0,+∞).

Of course the inequality (<ref>) yields

    |𝒯(t_0)|^2_H(t_0)(q)≤C̃_0.

So there exists a uniform constant C̅_1 depending only on C̅_0,d^-1 and the geometry of Ω̃ such that

    sup_(x,t)∈Ω×[0,+∞)|h^-1(t)∂_H_0h(t)|_H_0≤C̅_1.



Now we want to prove the key identity (<ref>), which plays an important role in the proof of Theorem <ref>.

Suppose (M,D,g,ν) is a compact affine Gauduchon manifold with non-empty smooth boundary ∂ M. Let ϕ be  a smooth function defined on M and satisfy the boundary condition ϕ|_∂ M=a, where a is a constant. Integrating by parts and applying Stokes' formula, one can obtain

    ∫_M|dϕ|^2ω^n_g/ν
    
      =   2n∫_M∂ϕ∧∂̅ϕ∧ω^n-1_g/ν
    
      =   2n∫_M∂(ϕ∂̅ϕ)∧ω^n-1_g/ν-2n∫_Mϕ∂∂̅ϕ∧ω^n-1_g/ν
    
      =   2n∫_M(a-ϕ)∂∂̅ϕ∧ω^n-1_g/ν-2n∫_M∂((a-ϕ)∂̅ϕ)∧ω^n-1_g/ν
    
      =   1/2∫_M(a-ϕ)Δ̃ϕω^n_g/ν+2n∫_M∂(∂̅(a-ϕ)^2∧ω^n-1_g)/ν+2n∫_M
      ∂̅(a-ϕ)^2∧∂ω^n-1_g/ν
    
      =   1/2∫_M(a-ϕ)Δ̃ϕω^n_g/ν+n∫_Md(∂̅(a-ϕ)^2∧ω^n-1_g/ν)+n∫_Md
      ((a-ϕ)^2∧∂ω^n-1_g/ν)
    
      =   1/2∫_M(a-ϕ)Δ̃ϕω^n_g/ν.

Then similar to the argument in <cit.>, we can get the following lemma.


  Assume (X,D,g,ν) is a special non-compact affine Gauduchon manifold admitting an exhaustion function ϕ with ∫_X|Δ̃ϕ|ω^n_g/ν<+∞, and suppose η is an (n-1)-form with ∫_X|η|^2ω^n_g/ν<+∞. Then if dη is integrable, we have
  
    ∫_Xdη=0.









































  Let (E,∇,θ) be a flat Higgs bundle with a fixed Hermitian metric H_0 over a special affine Gauduchon manifold (M,g,D,ν). Let H be a Hermitian metric on E and s:=log(H_0^-1H). If one of the following three conditions is satisfied:
  
    
  (1)Suppose that M is a compact manifold without boundary.
    
  (2)Suppose that M is a compact manifold with non-empty smooth boundary ∂ M, and H|_∂ M=H_0|_∂ M.
    
  (3)Suppose that M is a non-compact manifold admitting an exhaustion function ϕ with  ∫_M|Δ̃ϕ|ω^n_g/ν<+∞. Furthermore, we also assume that |∂ω^n-1_g/ν|_g∈ L^2(M),s∈ L^∞(M), and D_H_0,θ^1,0s∈ L^2(M), where D_H_0,θ^1,0=∂_H_0+θ^*H_0.
  
  Then, we have the following identity:
  
    ∫_M tr(Φ(H_0,θ)s)ω^n_g/ν+∫_M⟨Ψ(s)(D”s),D”s⟩_H_0ω^n_g/ν=∫_M tr(Φ(H,θ)s)ω^n_g/ν,

where D”=∂̅_E+θ and Ψ is the function defined by

    Ψ(x,y)={   e^y-x-1/y-x,     x≠ y,
       1,     x=y.
    .




  Set h=H_0^-1H=e^s. Obviously the definition tells us that
  
    tr((Φ(H,θ)-Φ(H_0,θ))s)=⟨ tr_g((h^-1∂_H_0h)+[θ,h^-1[θ^*H_0,h]]),s⟩_H_0.

Based on tr(h^-1(∂_H_0hs))= tr(s∂_H_0s), tr(s[θ^*H_0,s])=0 and ∂∂̅ω^n-1_g=0, we derive
  
    ∫_M⟨ tr_g(∂̅(h^-1∂_H_0h)),s⟩_H_0ω^n_g/ν
       =n∫_M tr(∂̅(h^-1∂_H_0h)s)∧ω^n-1_g/ν
       =n∫_M∂̅ tr(h^-1(∂_H_0h)s)∧ω^n-1_g/ν+n∫_M tr(h^-1∂_H_0h∧∂̅s)∧ω^n-1_g/ν
       =n∫_M tr(s∂_H_0s)∧∂̅ω^n-1_g/ν+n∫_M∂̅( tr(s∂_H_0s)∧ω^n-1_g)/ν
         +n∫_M tr(h^-1∂_H_0h∧∂̅s)∧ω^n-1_g/ν
       =n∫_M∂(1/2 tr(s^2)∧∂̅ω^n-1_g)/ν+n∫_M∂̅( tr(sD_H_0,θ^1,0s)∧ω^n-1_g)/ν
         +n∫_M tr(h^-1∂_H_0h∧∂̅s)∧ω^n-1_g/ν
       =n∫_Md(1/2 tr(s^2)∧∂̅ω^n-1_g/2ν)+(-1)^n-1∫_Md( tr(sD_H_0,θ^1,0s)∧ω^n-1_g/2ν)
          + n∫_M tr(h^-1∂_H_0h∧∂̅s)∧ω^n-1_g/ν.

  In condition (1), by using Stokes' formula; in condition (2), by using s|_∂ M=0 and Stokes' formula; in condition (3), by using lemma <ref>, we have
  
    ∫_M⟨ tr_g(∂̅(h^-1∂_H_0h)),s⟩_H_0ω^n_g/ν=∫_M tr(h^-1∂_H_0h∧∂̅s)∧ω^n-1_g/ν.

 According to <cit.>, we can also prove
 
    tr( tr_g(h^-1D_H_0,θ^1,0hD”s))=⟨Ψ(s)(D”s),D”s⟩_H_0

  and
  
    ∫_M tr( tr_g[θ,h^-1[θ^*H_0,h]]s)ω^n_g/ν=∫_M tr(h^-1[θ^*H_0,h][θ,s])∧ω^n-1_g/ν.

 From above, there holds that
  
    ∫_M⟨ tr_g(∂̅(h^-1∂_H_0h)+[θ,h^-1[θ^*H_0,h]]),s⟩_H_0ω^n_g/ν
      =∫_M⟨Ψ(s)(D”s),D”s⟩_H_0ω^n_g/ν.

  So (<ref>) immediately follows.




Next, we will consider the long-time existence of the affine Hermitian-Yang-Mills flow (<ref>) on some non-compact affine Gauduchon manifold (X,D,g) under some conditions of the initial metric H_0. In the following, we suppose that (X,D,g) satisfies the Assumption 2.

  Let {X_φ}_φ=1^∞ be an exhaustion sequence of compact sub-domains of X, i.e., they satisfy
  X_φ⊂ X_φ+1 and ⋃_φ=1^∞X_φ=X. We consider the  Dirichlet boundary condition:
  
    H|_∂ X_φ=H_0|_∂ X_φ.

  By Theorem <ref>, on every X_φ, there exists a unique long-time solution H_φ(t) of the following Dirichlet boundary problem:
  
    {   H_φ^-1(t)∂ H_φ(t)/∂ t=-4( tr_g(F_H_φ(t)+[θ,θ^*H_φ(t)])-λ· Id_E),
       H_φ(0)=H_0,
       H_φ(t)|_∂ X_φ=H_0|_∂ X_φ.
        .

  For simplicity, set Φ(H_φ(t),θ)= tr_g(F_H_φ(t)+[θ,θ^*H_φ(t)])-λ· Id_E. Suppose that there exists a positive number C_0 such that |Φ(H_0,θ)|_H_0≤ C_0 on X. Denote h_φ(t)=H_0^-1H_φ(t), then a direct calculation shows that
  
    |∂/∂ tlog trh_φ(t)|≤ 4|Φ(H_φ(t),θ)|_H_φ(t),

  and
  
    |∂/∂ tlog trh_φ^-1(t)|≤ 4|Φ(H_φ(t),θ)|_H_φ(t).

 Combining Proposition <ref> and the maximum principle, one can see
  
    sup_X_φ×[0,+∞)|Φ(H_φ,θ)|_H_φ≤ C_0.

  Integrating (<ref>) along the time direction from 0 to t, we get
  
    |log trh_φ(t)-log r|=|∫_0^t∂/∂ s(log trh_φ(s))ds|≤ 4C_0t.

  Then we have
  
    sup_X_φ×[0,T] trh_φ(t)≤ re^4C_0T,  inf_X_φ×[0,T] trh_φ(t)≥ re^-4C_0T,

  and
  
    sup_X_φ×[0,T] trh_φ^-1(t)≤ re^4C_0T,  inf_X_φ×[0,T]h_φ^-1(t)≥ re^-4C_0T.

  This implies that
  
    sup_X_φ×[0,T]σ(H_0,H_φ(t))≤ 2r(e^4C_0T-1).

  
    (<cit.>, <cit.>) Suppose u is a function on  X_φ×[0,T] satisfying
    
    (∂/∂ t-Δ̃)u≤ 0,   u|_t=0=0,

    and  there is a bound sup_X_φu≤ C_1. Then we have
    
    u(x,t)≤C_1/φ(φ(x)+C_2t),

    where C_2 is the bound of Δ̃ϕ in Assumption 2.
  

  In the following, we assume that there exists a constant C_0 such that
  
    sup_X|Φ(H_0,θ)|_H_0≤ C_0.


 Step1: We want to get the C^0-convergence of H_φ on any compact set Ω in finite time. For any compact subset Ω⊂ X, there exists a constant φ_0 such that Ω⊂ X_φ_0. Let H_φ(t) and H_ψ(t) be the long-time solutions of the flow (<ref>) on X_φ and X_ψ  for φ_0<φ<ψ. Let u=σ(H_φ,H_ψ). Of course(<ref>) gives a uniform bound of u and u satisfies (<ref>). By Lemma <ref>, we have
 
    σ(H_φ,H_ψ)≤C_1(φ_0+C_2T)/φ_0

  on X_φ_0×[0,T]. So H_φ is a Cauchy sequence on X_φ_0×[0,T] for φ→ +∞.

  Step2: Get the local C^∞-convergence of H_φ on any compact subset Ω. Clearly (<ref>) and Proposition <ref> give the uniform C^0 and local C^1 estimate of H_φ. One can get the local uniform C^∞-estimate of H_φ by the standard Schauder estimate of the parabolic equation. It should be pointed out that by applying the parabolic Schauder estimate, one can only get the uniform C^∞-estimate of h(t) on X_φ×[τ,T], where τ>0 and the uniform estimate depends on τ^-1. To fix this, one can use the maximum principle to get the local uniform bound on |F_H|_H, then apply the elliptic estimates to get the local uniform C^∞-estimates. We will omit here, because it is similar to <cit.>. By choosing a subsequence φ→ +∞, we know that H_φ(t) converge in C_ loc^∞-topology to a long-time solution H(t) of the heat flow (<ref>) on X.

  Let (X,D,g,ν) be a non-compact affine Gauduchon manifold satisfying the Assumptions 1 and 2, {X_φ} an exhausting sequence of compact sub-domains of X.
  By the definition, it is easy to check

    |D”s|_H_0^2≤C̃⟨Ψ(s)(D”s),D”s⟩_H_0,

where C̃ is a positive constant depending only on the L^∞-bound of s. By the identity (<ref>) in the compact and non-empty boundary case and the flow equation (<ref>), we have

    ∫_X_φ|D”s|_H_0^2ω^n_g/ν   ≤C̃∫_M⟨Ψ(s)(D”s),D”s⟩_H_0ω_g^n/ν
       =C̃∫_X_φ(- tr(Φ(H_0,θ)s)+ tr(Φ(H,θ)s))ω_g^n/ν
       =C̃(∫_X_φ(- tr(Φ(H_0,θ)s))ω_g^n/ν-1/8d/dt||s||_L^2^2)
       ≤C̃sup_X_φ|Φ(H_0,θ)|_H_0^2· Vol(X_φ,g)
       ≤ C(Φ(H_0,θ), Vol(X)).

  Since the right hand side of (<ref>) is independent of φ, we have
  
    ||D”(log(H_0^-1H))||_L^2≤ C(Φ(H_0,θ), Vol(X)).

   So we obtain the following theorem.

  
    Let (X,D,g) be a non-compact affine Gauduchon manifold satisfying the Assumption 1 and 2, (E,∇,θ) be a flat Higgs bundle with the background Hermitian metric H_0 over X. Suppose that there exists a positive number C_0 such that sup_X|Φ(H_0,θ)|_H_0≤ C_0<+∞, then the affine Hermitian-Yang-Mills flow (<ref>) has a long-time solution on X×[0,+∞). Furthermore, we also have
    
    ||D”(log(H_0^-1H))||_L^2≤ C(Φ(H_0,θ), Vol(X)).

 
 



 Let X be a non-compact  affine Gauduchon manifold satisfying the Assumption 1,2,3, and (E,∇,θ) be a flat Higgs bundle over X with a background Hermitian metric H_0 satisfying sup_X| tr_gF_H_0,θ|_H_0
<+∞.

  Let {X_φ}_φ=1^∞ be an exhaustion sequence of compact sub-domains of X. On every X_φ, there is a unique long-time solution H_φ(x,t) satisfies (<ref>).
  Due to Theorem <ref>, we know that H_φ(x,t) converge to the long-time solution H(x,t) of the affine Hermitian-Yang-Mills flow (<ref>) on X×[0,+∞) in C_ loc^∞-topology as φ→ +∞. We denote by A_0 the Chern connection on the flat vector bundle (E, ∇) with respect to the smooth  Hermitian metric H_0 and we sometimes denote A(t) by D_A(t). Let (A_φ(x,t),θ_φ(x,t)) be the long-time solution of the affine Yang-Mills-Higgs flow on the flat vector bundle (E, ∇, H_0) over the compact affine Gauduchon manifold (X_φ,D,g), i.e.
 
    {   ∂ A(t)/∂ t=-2(∂_A(t) tr_g-∂̅_A(t) tr_g)(F_A(t)+[θ(t),θ^*H_0(t)]),
       ∂θ(t)/∂ t=-2[ tr_g(F_A(t)+[θ(t),θ^*H_0(t)]),θ(t)],
       A(0)=A_0,  θ(0)=θ_0.
       .

 Following Donaldson (<cit.>) and Li-Zhang's (<cit.>) argument and setting σ(t)^*H_0∘σ(t)=h(t)=H_0^-1H(t), we can easily check that the affine Yang-Mills-Higgs flow (<ref>) is gauge equivalent to the affine Hermitian-Yang-Mills flow (<ref>) with the initial metric H_0. Here σ(t) is the gauge transformation, A(t)=σ(t)∘ D_H(t)∘σ^-1(t) and θ(t)=σ(t)∘θ_0∘σ^-1(t).

It is well known that
  
    |F_H(t)+[θ_0,θ_0^*H(t)]|^2_H(t)=|F_A(t)+[θ(t),θ(t)^*H_0]|^2_H_0,

  and
  
    |D_H(t)( tr_g(F_H(t)+[θ_0,θ_0^*H(t)]))|_H(t)=|D_A(t)( tr_g(F_A(t)+[θ(t),θ^*H_0(t)]))|_H_0.

  For simplicity, set
  
    Φ(H(t),θ_0)= tr_g(F_H(t)+[θ_0,θ_0^*H(t)])

  and
  
    Φ(A(t),θ(t))= tr_g(F_A(t)+[θ(t),θ^*H_0(t)]).

 Define
 
    I(t)=∫_X(|D_H(t)Φ(H(t),θ_0)|_H(t)^2+2|[Φ(H(t),θ_0),θ_0]|^2_H(t))ω^n_g/ν.

 Then using (<ref>) and (<ref>) we can easily check that
  
    I(t)=∫_X(|D_A(t)Φ(A(t),θ(t))|_H_0^2+2|[Φ(A(t),θ(t)),θ(t)]|^2_H_0)ω^n_g/ν.

 Now we want to prove that I(t)→ 0 as t→ +∞ by using the exhaustion method. Set

    I_φ(t)=∫_X_φ(|D_A_φ(t)Φ(A_φ(t),θ_φ(t))|_H_0^2+
    2|[Φ(A_φ(t),θ_φ(t)),θ_φ(t)]|^2_H_0)ω^n_g/ν.

According to the  uniform bound of sup_X |Φ(H_0,θ_0)|_H_0<+∞ and the maximum principle, we know that there exists a uniform constant C_0 such that

    sup_X_φ|Φ(A_φ(t),θ_φ(t))|_H_0=sup_X_φ|Φ(H_φ(t),θ_0)|_H_φ(t)<sup_X_φ|Φ(H_0,θ_0)|_H_0≤ C_0.

  
    Suppose X_φ is a special compact affine Gauduchon manifold with boundary ∂ X_φ and ||dω_g||_L^∞(X)<+∞. If A_φ(t) is a smooth solution of (<ref>), then it holds that
    
    dI_φ(t)/dt≤ĈI_φ(t)   for   t>t_0>0,

    where Ĉ=Ĉ(n,r,C_0) is a uniform constant independent of t and φ.
  
  
    For simplicity, we denote A by A_φ(t) and θ by θ_φ(t). And  we will omit H_0 if we know that the adjoint is with respect to H_0. By direct calculations, one can check
    
    ∂/∂ tΦ   = tr_g(D_A(∂ A/∂ t)+[∂θ/∂ t,θ^*]+[θ,(∂θ/∂ t)^*])
       =-2 tr_gD_A(∂_A-∂̅_A)Φ+2 tr_g([θ,[Φ,θ^*]]-[[Φ,θ],θ^*]),

  and
  
    dI_φ(t)/dt   =∫_X_φ∂/∂ t(|D_AΦ|^2+2|[Φ,θ]|^2)ω_g^n/ν
       =2 Re∫_X_φ⟨∂/∂ t(D_AΦ),D_AΦ⟩+2⟨∂/∂ t[Φ,θ],[Φ,θ]⟩ω_g^n/ν
       =2 Re∫_X_φ⟨[∂ A/∂ t,Φ]+D_A(∂Φ/∂ t),D_AΦ⟩ω_g^n/ν
          +4 Re∫_X_φ⟨ [∂Φ/∂ t,θ]+[Φ,∂θ/∂ t],[Φ,θ]⟩ω_g^n/ν.

   By the parabolic equation and the Dirichlet boundary condition (<ref>), we have
   
    Φ(H_φ(t),θ_0)|_∂ X_φ=0   for   t>t_0>0.

 After a gauge transformation, we also see Φ(A_φ(t),θ_φ(t))|_∂ X_φ=0. So for t>t_0>0, we have
  
    ∂/∂ tΦ(A_φ,θ_φ)|_∂ X_φ=0.

  Setting l=[Φ,θ], we have
 
    ⟨ [ tr_g[-l,θ^*],θ],l⟩   =-⟨[g^αβ(l_αθ^*_β-θ_β^*l_α),θ],l⟩
       =-g^αβg^γη tr(((l_αθ_β^*-θ_β^*l_α)θ_γ-θ_γ
       (l_αθ_β^*-θ_β^*l_α))l_η^*)
       =- tr((g^αβ(l_αθ_β^*-θ_β^*l_α))(g^ηγ(l_ηθ_γ^*-θ^*_γl_
       η))^*)=-| tr_g[l,θ^*]|^2

 and

    |[Φ,θ]|^2   =⟨ l,Φθ-θΦ⟩=g^αβ tr(l_α(Φθ_β)^*-l_α(θ_βΦ)^*)
       = tr(g^αβ(l_αθ_β^*-θ_β^*l_α)Φ^*)=⟨ tr_g[l,θ^*],Φ⟩
       ≤ |Φ|| tr_g[l,θ^*]|.

Applying the Stokes' formula again and substituting (<ref>) , we obtain

    2 Re∫_X_φ⟨[∂ A/∂ t,Φ]+D_A(∂Φ/∂ t),D_AΦ⟩ω^n_g/ν
    
    =   2 Re∫_X_φ(⟨[2(∂̅_A-∂_A)Φ,Φ],D_AΦ⟩+
      ⟨∂/∂ tΦ,D_A^*D_AΦ⟩)ω^n_g/ν
    
    =   4 Re∫_X_φ⟨[(∂̅_A-∂_A)Φ,Φ],D_AΦ⟩ω^n_g/ν
      -4∫_X_φ|D_A^*D_AΦ|^2ω^n_g/ν
        -4 Re∫_X_φ⟨(τ + τ̅)^*D_AΦ,D_A^*D_AΦ⟩ω^n_g/ν
        +4 Re∫_X_φ⟨ tr_g([θ,[Φ,θ^*]]-[[Φ,θ],θ^*]),D_AΦ⟩ω^n_g/ν,

where τ=[ tr_g,∂ω_g].

Combining (<ref>), (<ref>) and (<ref>), we deduce

    4 Re∫_X_φ⟨[∂Φ/∂ t, θ],[Φ,θ]⟩ω^n_g/ν
       =8 Re∫_X_φ⟨[-D_A^*D_AΦ,θ],[Φ,θ]⟩ω^n_g/ν-8 Re∫_X_φ⟨[(τ+τ̅)^*D_AΦ,θ],[Φ,θ]⟩ω^n_g/ν
          +8 Re∫_X_φ⟨[ tr_g[θ,(-[Φ,θ])^*],θ],[Φ,θ]⟩ω^n_g/ν
          +8  Re∫_X_φ⟨[ tr_g[-[Φ,θ],θ^*],θ],[Φ,θ]⟩ω^n_g/ν
       ≤ 16∫_X_φ|θ||[Φ,θ]||D_A^*D_AΦ|ω^n_g/ν+16C_1
      ∫_X_φ|θ||[Φ,θ]||D_AΦ|ω^n_g/ν
          +C_2∫_X_φ|θ|^2|[Φ,θ]|^2ω^g/ν-8∫_X_φ| tr_g[l,θ^*]|^2ω^n_g/ν

and

    4 Re∫_X_φ⟨[Φ,∂θ/∂ t],[Φ,θ]⟩ω^n_g/ν
       =-8  Re∫_X_φ⟨[Φ,[Φ,θ]],[Φ,θ]⟩ω_g^n/ν
       ≤ 16∫_X_φ|Φ||[Φ,θ]|^2ω^n_g/ν,

where C_1 is a uniform constant determined by ||dω_g||_L^∞(X)<+∞ and C_2=C_2(n,r) only depends on n and r.

Putting (<ref>)-(<ref>) into (<ref>), we have

    dI_φ/dt≤    C_2∫_X_φ(|Φ||D_AΦ|^2+(|Φ|+|θ|^2)|[Φ,θ]|^2+|θ||[Φ,θ]|
      |D_A^*D_AΦ|)ω^n_g/ν
       -4∫_X_φ|D_A^*D_AΦ|ω^n_g/ν-8∫_X_φ| tr_g[l,θ^*]|^2ω_g^n/ν
        +16(C_1+1)∫_X_φ|θ||[Φ,θ]||D_AΦ|ω_g^n/ν
      +4C_1∫_X_φ|D_AΦ||D_A^*D_AΦ|ω^n_g/ν
    ≤   ĈI_φ(t),

where Ĉ=Ĉ(n,r,C_0) is a uniform constant independent of t and φ.
 

 
   Let H(t) be the long-time solution of the affine Hermitian-Yang-Mills flow (<ref>) with the initial metric H_0, which is constructed in Theorem <ref>. Then I(t)→ 0 as t→ +∞.
 


 By (<ref>) in Lemma <ref>, we know that there exists a uniform constant Ĉ such that
 
    I_φ(t)≤ e^Ĉ(t-s)I_φ(s)

for any 0<t_0≤ s≤ t.

 On the other hand, there holds that
 
    ∫_X_φ|Φ(A_φ(t),θ_φ(t))|_H_0^2ω^n_g/ν+2∫_t_0^t∫_X_φ|D_A_φΦ(A_φ,θ_φ)|_H_0^2ω^n_g/νds
    
     =   ∫_X_φ|Φ(A_φ(t_0),θ_φ(t_0))|_H_0^2ω^n_g/ν.

 According to Fatou's Lemma, we get
 
    ∫_X|Φ(A(t),θ(t))|_H_0^2ω^n_g/ν+2∫_t_0^tlim inf_φ→∞∫_X_φ|D_A_φΦ(A_φ,θ_φ)|_H_0^2ω^n_g/νds
    ≤   ∫_X|Φ(A(t_0),θ(t_0))|_H_0^2ω^n_g/ν.

 This implies that ∫_X|Φ(A(t),θ(t))|_H_0^2ω^n_g/ν  is monotonically
nonincreasing with respect to t. Then we have

    (∫_X|Φ(A(t),θ(t))|_H_0^2ω^n_g/ν-∫_X|Φ(A(t+1),θ(t+1))|_H_0^2ω^n_g/ν)→ 0,     as   t →∞.

For any m≥ t_0>0, there exists t_m∈ [m,m+1] such that

    I_φ(t_m)=∫_m^m+1I_φ(t)dt.

From (<ref>),(<ref>) and (<ref>), it follows that
 
    I_φ(t)≤    e^2ĈI_φ(t_m)=e^2Ĉ∫_m^m+1I_φ(t)dt
    
      =   e^2Ĉ/2(∫_X_φ|Φ(A_φ(m),θ_φ(m))|_H_0^2ω^n_g/ν-∫_X_φ|Φ(A_
      φ(m+1),θ_φ(m+1))|_H_0^2ω^n_g/ν)
       +e^2Ĉ∫_m^m+1∫_X_φ|[Φ(A_φ,θ_φ),θ_φ]|^2ω^n_g/νdt

for any t∈ [m+1,m+2]. Applying Fatou's Lemma again, we drive

    I(t)≤   lim inf_φ→∞I_φ(t)
    ≤   lim inf_φ→∞e^2Ĉ/2(∫_X_φ|Φ(A_φ(m),θ_φ(m))|_H_0^2ω^n_g/ν-∫_X_φ|Φ(A_
      φ(m+1),θ_φ(m+1))|_H_0^2ω^n_g/ν)
    
      =   e^2Ĉ/2(∫_X|Φ(A(m),θ(m))|_H_0^2ω^n_g/ν-∫_X|Φ(A(m+1),θ(m+1))
      |_H_0^2)
       +e^2Ĉ∫_m^m+1∫_X_φ|[Φ(A_φ,θ_φ),θ_φ]|^2ω^n_g/νdt

for any t∈ [m+1,m+2]. Together with (<ref>), it means that I(t)→ 0 as t→ +∞.





§ STABLE CASE


Let (X,D,g,ν) be a non-compact special affine Gauduchon manifold satisfying the Assumptions 1,2,3, and |dω^n-1_g|_g∈ L^∞(X), (E,∇,θ) be a flat Higgs bundle over X. Fix a proper background Hermitian metric H_0 satisfying sup_X| tr_gF_H_0,θ|_H_0<+∞ on E. According to <cit.>, we can solve the following Poisson equation on (X,D,g,ν):

    tr_g∂̅∂ f=-1/r tr( tr_gF_H_0,θ-λ· Id_E),

where

    λ=∫_X tr( tr_gF_H_0,θ)ω^n_g/ν/ rank(E) Vol(X).

By a conformal change H̅_0=e^fH_0, we can check that H̅_0 satisfies

    tr( tr_g(F_H̅_0+[θ,θ^*H̅_0])-λ· Id_E)=0.

So without loss of generality, we can assume that the initial metric H_0 satisfies the condition (<ref>). Let H(t) be the long-time solution of the affine Hermitian-Yang-Mills flow (<ref>) with the initial metric H_0. Set h(t)=H_0^-1H(t)=e^s(t). Then it follows that

    log(h(t))= tr(s(t))=0.



Suppose X is a non-compact special affine Gauduchon manifold as before, then there are constants C_1 and C_2 such that

    sup_X|s(t)|≤ C_1 ||s(t)||_L^2(X)+C_2.



From  <cit.>, we have
  
    Δ̃log( trh(t)+ trh^-1(t))≥ -2(| tr_gF_H(t),θ|_H(t)+| tr_gF_H_0|_H_0).

  On the basis of Proposition <ref>, it is easy to see that | tr_gF_H(t),θ|_H(t) is uniformly bounded. On the other hand, we know
  
    log(1/2r( trh(t)+ trh^-1(t)))≤ |log h(t)|≤ r^1/2log( trh(t)+ trh^-1(t)).

  Then  Assumption 3 implies (<ref>).


Proof of Theorem <ref>

When (E,∇,θ) is stable, we will show that, by choosing a subsequence, H(t_i) converge to an affine Hermitian-Einstein metric H_∞ in C_loc^∞ as t_i→ +∞. Clearly Proposition <ref> and  the standard elliptic estimates mean that we only need to obtain a uniform C^0-estimate. By Lemma <ref>, the key is to get a uniform L^2-estimate for log h(t), i.e.  there exists a constant Ĉ independent of t such that

    ||log h(t)||_L^2=(∫_X|log h(t)|^2_H(t)ω^n_g/ν)^1/2≤Ĉ

for all t>0. We prove (<ref>) by contraction. If not, there would exist a subsequence t_i→ +∞, such that

    lim_i→ +∞||s(t_i)||_L^2=+∞

and

    _i→ +∞d/dt||s(t)||_L^2|_t=t_i≥ 0.

In fact, there must exist a subsequence t̂_i→ +∞, such that

    lim_i→ +∞||s(t̂_i)||_L^2=+∞

and

    ||s(t̂_i)||_L^2<||s(t̂_i+1)||_L^2

for all t̂_i<t̂_i+1. Denote t̃_i=max{t∈ [t̂_i,t̂_i+1] |   ||s(t)||_L^2≤ ||s(t̂_i)||_L^2}, it is easy to see that t̃_̃ĩ<t̂_i+1, then there exist t_i∈[t̃_i,t̂_i+1] such that

    d/dt||s(t)||_L^2|_t=t_i=||s(t̂_i+1)||_L^2-||s(t̃_i)||_L^2/t̂_i+1-t̃_i
    >0

and ||s(t_i)||_L^2≥||s(t̂_i)||_L^2. So we obtain a sequence t_i ∈ (t̂_i,t̂_i+1) such that ||s(t_i)||_L^2>||s(t̂_i)||_L^2 and (d/dt||s(t)||_L^2)|_t=t_i>0.
Set

    s(t_i)=log h(t_i),   l_i=||s(t_i)||_L^2,   u_i=s(t_i)/l_i.

Then tr(u_i)=0,   ||u_i||_L^2=1, and

    sup|u_i|≤1/l_i(C_1l_i+C_2)<C_3<+∞.


∙ Step 1: We show that u_i converge to u_∞  in L_1^2 weakly as i→ +∞. We need to show that ||u_i||_L_1^2 are uniformly bounded. Since ||u_i||_L^2=1, it is enough to prove ||D”u_i||_L^2 are uniformly bounded.

Noting (<ref>) and applying to the flow equation (<ref>), one can obtain

    ∫_X tr(Φ(H_0,θ)s(t))+⟨Ψ(s(t))(D”s(t)),D”s(t)⟩_H_0ω^n/ν
    
      =   ∫_X tr(Φ(H,θ)s(t))ω^n_g/ν
      =-1/4∫_X tr(H^-1(t)∂ H(t)/∂ ts(t))ω^n_g/ν
    
      =   -1/4∫_X tr(∂ s(t)/∂ ts(t))ω^n_g/ν
      =-1/8d/dt∫_X tr(s^2(t))ω^n_g/ν
    
      =   -1/8d/dt||s(t)||_L^2^2.

Then

    ∫_X tr(Φ(H_0,θ)u_i)+l_i⟨Ψ(l_iu_i)(D”u_i),D”u_i⟩_H_0ω^n_g/ν=-1/4d/dt||s(t_i)||_L^2<0.

Consider the function

    lΨ(lx,ly)={   l,     x=y,
       e^l(y-x)-1/y-x,     x≠ y.
    .

Because of (<ref>), we may assume that (x,y)∈ [C_3,C_3]×[-C_3,C_3]. It is easy to check that

    lΨ(lx,ly)→{   1/x-y,   x>y,
       +∞,    x≤ y,
    .

increases monotonically as l → +∞. Let ζ∈ C^∞(ℝ×ℝ,ℝ^+) satisfy ζ(x,y)<1/x-y whenever x>y. Together with (<ref>), (<ref>) gives us that

    ∫_X tr(Φ(H_0,θ)u_i)ω^n_g/ν+∫_X⟨ζ(u_i)(D”u_i),D”u_i⟩_H_0ω^n_g/ν≤ 0,     i≫ 0.

In particular, take ζ(x,y)=1/3C_3. It is obvious that when (x,y)∈ [-C_3,C_3]×[-C_3,C_3] and x>y, 1/3C_3<1/x-y. This implies

    ∫_X tr(Φ(H_0,θ)u_i)ω^n_g/ν+1/3C_3∫_X|D”u_i|_H_0^2ω^n_g/ν≤ 0,

for i≫ 0. Then

    ∫_X|D”u_i|_H_0^2ω^n_g/ν≤ 3C_3^2sup_X|Φ(H_0,θ)|_H_0 Vol(X).

Thus, u_i are bounded in L_1^2. We can choose a subsequence {u_i_j} such that u_i_j⇀ u_∞ weakly in L_1^2, still denoted by {u_i}_i=1^∞ for simplicity. Noting that L_1^2↪ L^2, we have

    1=∫_X|u_i|_H_0^2ω^n_g/ν→∫_X|u_∞|_H_0^2ω^n_g/ν,   as   i→ +∞.

This indicates that ||u_∞||_L^2=1 and u_∞ is non-trivial. Moreover, tru_∞=0. Using (<ref>) and following a similar discussion as that in <cit.>, we deduce

    ∫_X tr(Φ(H_0,θ)u_∞)ω^n_g/ν+∫_X⟨ζ(u_∞)(D”u_∞),D”u_∞⟩_H_0ω^n_g/ν≤ 0.



∙ Step 2 Using Uhlenbeck and Yau's trick <cit.> and Loftin's argument <cit.>, we can construct a  flat Higgs sub-bundle which contradicts the stability of (E,∇,θ).

By (<ref>)  and the same argument  in <cit.>, we know that the eigenvalues of u_∞ are constant almost everywhere. Let μ_1<μ_2<⋯<μ_l be the distinct eigenvalues of u_∞. The fact that tr(u_∞)=0 and ||u_∞||_L^2=1 forces 2≤ l≤ r. For each μ_α(1≤α≤ l-1), we construct a function P_α:ℝ→ℝ such that

    P_α={
      1,     x≤μ_α,
    
      0,     x≥μ_α +1.
    .

Setting π_α=P_α(u_∞), by <cit.>, we have

  
  (i)π_α∈ L_1^2;
  
  (ii)π_α^2=π_α=π_α^*H_0;
  
  (iii)( Id_E-π_α)∂̅π_α=0;
  
  (iv)( Id_E-π_α)[θ, π_α]=0,

and {π_α}_α=1^l-1 determine (l-1) flat Higgs sub-bundles of E. Set E_α=π_α(E). From tr(u_∞)=0 and u_∞=μ_l· Id_E-∑_α=1^l-1(μ_α+1-μ_α)π_α, it holds that

    μ_l· rank(E)=∑_α=1^l-1(μ_α+1-μ_α) rank(E).

Construct

    γ=μ_l deg(E,H_0)-∑_α=1^l-1(μ_α+1-μ_α) deg(E_α,H_0).

Substituting (<ref>) into γ, we obtain

    γ=∑_α=1^l-1(μ_α+1-μ_α) rank(E_α)( deg(E,H_0)/ rank(E)- deg(E_α,H_0)/ rank(E_α)).

Applying the same argument as that in <cit.>, by the Chern-Weil formula, we have

    deg(E_α,H_0)=1/n∫_X( tr(π_α tr_gF_H_0,θ)-|D”π_α|^2)ω^n_g/ν.

On the other hand, putting (<ref>) into γ, we have

    γ=   μ_l/n∫_X tr( tr_gF_H_0,θ)ω^n_g/ν-1/n∑_α=1^l-1(μ_α+1-μ_α)
      ∫_X( tr(π_α tr_gF_H_0,θ)-|D”π_α|^2)ω^n_g/ν
    
      =   1/n∫_X tr((μ_l· Id_E-∑_α=1^l-1(μ_α+1-μ_α)π_α)( tr_gF_H_0,θ))ω^n_g/ν
       +1/n∑_α=1^l-1(μ_α+1-μ_α)∫_X|D”π_α|^2ω^n_g/ν
    
      =   1/n∫_X( tr(u_∞ tr_gF_H_0,θ)+⟨∑_α=1^l-1(μ_α+1-μ_α)(dP_α)^2(u_∞)(D”u_∞),
      D”u_∞⟩_H_0)ω^n_g/ν,

where the function dP_α:ℝ×ℝ→ℝ is defined by

    dP_α(x,y)={   P_α(x)-P_α(y)/x-y,   x≠ y,
       P_α'(x),   x=y.
    .

One can easily check that

    ∑_α=1^l-1(μ_α+1-μ_α)(dP_α)^2(μ_β-μ_γ)=
    |μ_β-μ_γ|^-1,   if  μ_β≠μ_γ.

 Using (<ref>), we derive

    γ   =1/n∫_X( tr(u_∞ tr_gF_H_0,θ)+⟨∑_α=1^l-1(μ_α+1-μ_α)(dP_α)^2(u_∞)(D”u_∞),
      D”u_∞⟩_H_0)ω^n_g/ν
       ≤ 0.

Then (<ref>) means

    ∑_α=1^l-1(μ_α+1-μ_α) rank(E_α)( deg(E,H_0)/ rank(E)- deg(E_α,H_0)/ rank(E_α))≤ 0,

which contradicts the stability of E.



Since we have proved that H(t_i) converge to the metric H_∞ in C_ loc^∞ as t_i → +∞, it remains for us to show that the limit metric H_∞ is an affine Hermitian-Einstein metric.
We will prove it by using I(t)→ 0 as  t → +∞.

Firstly, we can easily show that (E,H_0)=(E,H_∞) under above assumptions. Indeed, set h_∞=H_0^-1H_∞, then we

    (E,H_∞)-(E,H_0)=∫_X∂̅∂log (h_∞)∧ω_g^n-1/ν
       =∫_X∂̅(∂log (h_∞)∧ω_g^n-1)/ν+
      ∂(log (h_∞)·∂̅ω_g^n-1)/ν-log (h_∞)·∂∂̅ω_g^n-1/ν
       =(-1)^n-1∫_Xd(∂log (h_∞)∧ω_g^n-1/2ν)+∫_Xd(log (h_∞)·∂̅ω_g^n-1/2ν)
       =0,

where we have used Lemma <ref> to show that the last two terms vanish. The first term vanishes because  ||D”(log(H_0^-1H(t)))||_L^2 is uniformly bounded and the second vanishes because |dω_g^n-1|∈ L^∞(X) and |log (h_∞)|∈ L^∞(X).

From I(t)→ 0 as t→ +∞, we know that

    {
        D_H_∞Φ_∞=0,
    
        [Φ_∞,θ_∞]=0,
      .

where Φ_∞=Φ(H_∞,θ_∞). Since Φ_∞ is parallel and Φ_∞^*H_∞=Φ_∞, we can decompose (E_∞,H_∞,θ_∞) according to the real eigenvalues λ_1⋯λ_k of Φ_∞, i.e.

    (E_∞,H_∞,θ_∞)=⊕_i=1^k (E^i_∞,H_∞|_E_∞^i,θ_∞|_E^i_∞).

Set H^i=H_∞|_E^i_∞ and Φ_i=Φ_∞|E^i_∞, then we have

    Φ_i=λ_i  Id_E^i_∞.

If k=1, we are done. Otherwise,  it contradicts the H_0-stability of E. In fact, since E^i_∞ is mutually orthogonal flat sub-bundle of E with respect to H_∞, we have

    (E,H_0)=∑_i=1^k(E^i_∞,H^i).

So there exists  an i_0 such that μ_g(E,H_0)≤μ_g(E_∞^i_0,H^i_0). Then a contradiction occurs.



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