Changpeng PAN

Abstract In this paper,the author solves the Dirichlet problem for Hermitian-Poisson metric equation=λId and proves the existence of Hermitian-Poisson metrics on flat bundles over a class of complete Hermitian manifolds.When λ=0,the Hermitian-Poisson metric is a Hermitian harmonic metric.

Keywords Flat bundle,Hermitian harmonic metric,Hermitian-poisson metric,Complete Hermitian manifolds

1 Introduction

Let(X,g)be a Hermitian manifold,ω be the Khler form related to g.Let(V,D)be a flat bundle of rank r over X,i.e.,the connection satisfies D2=0.For any Hermitian metric H on V,we have the following unique decomposition:

where DHis compatible with H and ψH∈(End(E))is self-adjoint with respect to H.Set

Define

and GH=2.The harmonic metric equation is

We say H is a harmonic metric if it satisfies the harmonic metric equation.When(X,g)is a Khler manifold,Khler identity implies thatSo the harmonic metric equation is equivalent to

It is well known that there is a correspondence between flat bundles and representations of fundamental group.Let ρ:π1(X)→GL(r,C)be the representation related to(V,D).The Hermitian metric H induces a ρ-equivariant map

When the rank of the bundle equals to 1,(1.4)becomes a Poisson equation.For the high rank case,Donaldson[4]and Corlette[2]proved the existence of harmonic metrics on semisimple flat bundles over compact Riemannian manifolds.Jost-Zuo proved the existence of harmonic metrics on semi-simple flat bundles over quasi-compact Khler manifolds in[8–9].In[1],Collins,Jacob and Yau considered the following Poisson metric equation over non-compact curves:

This is a deformation of the harmonic metric equation.They proved the existence of Poisson metric on polystable parabolic flat bundle.

In this paper,we are interested in the following Hermitian-Poisson metric equation:

over complete Hermitian manifolds.We call H on V is a Hermitian-Poisson metric if H satisfies(1.7).When λ=0,we call H is a Hermitian harmonic metric.

We first prove the Dirichlet problem for Hermitian-Poisson metric equation over compact Hermitian manifolds with smooth boundary.

Theorem 1.1Assume that(X,g)is a compact Hermitian manifold with non-empty smooth boundary∂X.Let(V,D)be a flat bundle over X.Then there is a unique Hermitian-Poisson metric H on X such that H|∂X=ϕ,where ϕ is a Hermitian metric on V|∂X.

For a non-compact complete Hermitian manifold(X,g),take a compact sub-domains exhausting sequenceof X.Then the Poisson metric equation can be solved on Ωifor every i.Suppose that the manifold(X,g)satisfies some suitable conditions and there exists a good background Hermitian metric on V.Then we can deform these Poisson metrics on Ωiinto a Poisson metric on X.

Remark 1.1Suppose that X is a Khler manifold and λ=0.Then Theorem 1.1 is a special case of Dirichlet problem for harmonic map equation from compact manifolds with smooth boundary to complete Riemannian manifold with nonpositive sectional curvature.Theorem 1.2 should be a special case of harmonic map equation from complete noncompact manifolds to complete Riemannian manifold with nonpositive sectional curvature.See[3]for detail(see[7,10]for Hermitian harmonic map).

This paper is organized as follows.In Section 2,we introduce a heat flow about Poisson metric equation and prove the long time existence of its solution.In Section 3,we prove Theorem 1.1,and in Section 4,we prove Theorem 1.2.

2 Heat Flow on Compact Manifolds with Boundary

where H0is a background metric on V.It is not hard to check that this is a nonlinear parabolic equation,so the solution exists for short time.

Proposition 2.1Let H(t)be the solution of(2.1).Then

ProofWhen there is no confusion,we omit the parameter t in the computations for simplicity.Under the local flat basis of(V,D),we have

A direct computation implies

and

So

where in the last equality we have used the fact thatOn the other hand,notice that Φ(H)*H=Φ(H),so

Combining all the above,we have

On the boundary∂X we know Φ(H(t))=0.By the maximum principle,(2.3)holds.

Definition 2.1For any two Hermitian metrics H and K on V,define

Let h=K−1H andThen we find

Lemma 2.1Let H and K be two Hermitian-Poisson metrics.Then we have

ProofFrom(2.11),one can see that

Taking trace of both sides,we get

Also,we can derive

Since Φ(H)=Φ(K)=0,this lemma follows.

Lemma 2.2Let H(t)and K(t)be two solutions of(2.1).Then

ProofLet h(t)=K−1(t)H(t).Note that

and

This finishes the proof.

We will show the long-time existence of the solution in the following.

Proposition 2.2If H(t)is a solution of the parabolic equation(2.1)defined for 0≤t

ProofGiven∊>0,by the continuity at t=0 we can find a δ such that

for 0

for all s,t>T−δ.Then H(t)are uniform Cauchy sequence and converge to a continuous limiting metric HT.Set h(t)=K−1H(t).A direct calculation shows

and

This together with Proposition 2.1 means that HTis non-degenerate.

Following Simpson’s argument(see[12,Lemma 6.4]),we can conclude the following lemma.

Lemma 2.3Suppose that H(t)is a family of metrics on V over X with H(t)→HTin C0-norm.If H(t)satisfy Dirichlet boundary conditions,and if supX|ΛωGH(t)|H(t)is bounded uniformly in t,then H(t)are bounded inuniformly in t.

Corollary 2.1The parabolic equation(2.1)has a unique solution H(t)which exists for 0≤t<+∞.

3 Proof of Theorem 1.1

In this section we will consider the Dirichlet boundary problem for Hermitian-Poisson metric equation and use the heat equation method to deform an arbitrary initial metric to the desired one.The main points in the discussion are similar to that in[5]or[12].Let X be a compact Hermitian manifolds with smooth boundary∂X.For any Hermitian metric ϕ on V|∂Xover∂X.We can extend it to V over X,denoted by H0.Let H(t)be the solution of(2.1).

Proof of Theorem 1.1By a direct calculation,one can check that

for any section γ∈Γ(X,V).According to Proposition 2.1,we have

Let v be a solution of the following equation(see[14,Chapter 5,Proposition 1.8]for the existence of v):

The maximum principle implies

for any 0≤t<+∞.Let 0≤t1≤t and=H−1(t1)H(t).Then

and

From the above formula,we see

We have a similar estimate for tr(H−1(t)H(t1)).This gives us that

Combining(3.5)and(3.9),we deduce that H(t)converge in the C0topology to some continuous metric H∞as t→+∞.Using Lemma 2.3 again,one can find that H(t)are bounded inuniformly in t.By the heat equation,is bounded.Then,the standard elliptic regularity implies that there exists a subsequence H(t)→H∞in C∞topology.Due to formula(3.5),we know that H∞satisfies

From Lemma 2.1 and the maximum principle,it is easy to conclude the uniqueness of solution.

4 Proof of Theorem 1.2

In this section,we study the existence of the Hermitian-Poisson metrics on some complete Hermitian manifolds.The argument is similar to that used by Zhang in[15](also in[11,16]).

Suppose that X is a complete noncompact Hermitian manifold.Letbe an exhausting sequence of compact sub-domains of X,and K be a Hermitian metric on V.In Section 3,we have already shown that the following Dirichlet problem is solvable on Ωi,i.e.,there exists a Hermitian metric Hisuch that

In order to prove that we can pass to limit and eventually obtain a solution on the whole manifold X,we need to establish some estimates.Denote hi=K−1Hiand(Hi,K)=log(tr(hi)+tr()−log(2r).Then we have

Definition 4.1We say thathas the positive first eigenvalue if there exists a constant c>0 such that for any smooth function ρ with compact support,one has

And the supremum of c is denoted by.

Definition 4.2We say thatsatisfies L2-Sobolev inequality if there exists a constant S(X)such that for any smooth function ρ with compact support,one has

Lemma 4.1(see[15])Let X be an n-dimensional complete Hermitian manifold,and the holomorphic Laplace operatorhas the positive first eigenvalue(X).Then for a nonnegative continuous function f,the equation

has a nonnegative solution u∈(0<α<1)if f∈Lp(X)for some p≥2.

Lemma 4.2(see[15])Let X be an n-dimensional complete Hermitian manifold,and the holomorphic Laplace operatorsatisfies the L2-Sobolev inequality.Then for a nonnegative continuous function f,the equation

has a nonnegative solution u∈(0<α<1)if f∈Lp(X)for some n>p≥2.

Proof of Theorem 1.2(i)By Lemma 4.1(Lemma 4.2 for(ii)),(4.2)and the maximum principle,we conclude that

on Ωi,and u is the solution of=−4|Φ(K)|K.After a similar argument with[15],we can show that the C1-norm of Hiare uniformly bounded on any bounded open subset.Then the standard elliptic theory tells us that,by passing a subsequence,Hiconverge uniformly on any compact sub-domain of X to a smooth Hermitian metric H∞satisfying

AcknowledgementThe author would like to express his deep gratitude to Prof.Xi Zhang for numerous help and valuable guidance.