the poisson equation on complete manifolds with positive spectrum and applications

Advances in Mathematics 223 (2010) 198–219www.elsevier.com/locate/aim

The Poisson equation on complete manifoldswith positive spectrum and applications

Ovidiu Munteanu a, Natasa Sesum b,∗,1

a Columbia University, New York, NY 10027, United Statesb University of Pennsylvania, Philadelphia, PA 19104, United States

Received 9 December 2008; accepted 10 August 2009

Available online 4 September 2009

Communicated by Gang Tian

Abstract

In this paper we investigate the existence of a solution to the Poisson equation on complete manifoldswith positive spectrum and Ricci curvature bounded from below. We show that if a function f has decayf = O(r−1−ε) for some ε > 0, where r is the distance function to a fixed point, then the Poisson equation�u = f has a solution u with at most exponential growth.

We apply this result on the Poisson equation to study the existence of harmonic maps between completemanifolds and also existence of Hermitian–Einstein metrics on holomorphic vector bundles over completemanifolds, thus extending some results of Li–Tam and Ni.

Assuming moreover that the manifold is simply connected and of Ricci curvature between two negativeconstants, we can prove that in fact the Poisson equation has a bounded solution and we apply this result tothe Ricci flow on complete surfaces.© 2009 Elsevier Inc. All rights reserved.

Keywords: Poisson equation; Complete manifold; Harmonic map; Hermitian–Einstein metric

1. Introduction

We consider a complete noncompact manifold M with a Riemannian metric g on it. Weassume that M has a positive spectrum, which means that λ1(M) > 0, where λ1(M) denotes the

* Corresponding author.E-mail addresses: [email protected] (O. Munteanu), [email protected] (N. Sesum).

1 Partially supported by the NSF grant DMS-06-04657.

0001-8708/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2009.08.003

O. Munteanu, N. Sesum / Advances in Mathematics 223 (2010) 198–219 199

greatest lower bound of the L2 spectrum of the Laplace operator on M . The second condition isthat the Ricci curvature is bounded from below by a negative constant.

In the first part of the paper we study the Poisson equation on (M,g),

�u = f, (1)

where f is a function on M with a decay

∣∣f (x)∣∣ = O

(d(p,x)−1−ε

),

with d(p,x) being the distance function on M with respect to a fixed point p ∈ M and ε > 0.The Poisson equation on complete manifolds has been an extensive subject of study for many

people over the years. The case of complete manifolds with nonnegative Ricci curvature has beenmore approachable, and in fact Ni, Shi and Tam gave in [21] necessary and sufficient conditionsfor existence of solutions with certain growth rates. Of great importance in their study were theestimates of Green’s function proved by Li and Yau [13]. In general though, when the assump-tion that Ricci curvature is nonnegative is removed, one does not have such estimates and theproblem becomes considerably more difficult. Results that we know assume either some type ofintegrability for the function f on the right hand side of (1), or that the manifold M has a certainstructure at infinity. For example, it is known that a solution of the Poisson equation exists ifthe manifold M has positive spectrum and the right hand side of the equation is in Lp(M) forsome 1 < p < ∞, see [19,25]. Interesting results about the Poisson equation can also be foundin the context of geometric scattering theory, see e.g. [15,16]. However, this applies only whenthe manifold has controlled asymptotic geometry.

Recall that our setting in this paper is that M has positive spectrum and Ricci curvaturebounded below. It is well known that manifolds with positive spectrum have exponential vol-ume growth, see [2,11], therefore assumptions on the integrability of f are quite restrictive.

Our main result in this paper is that we can solve the Poisson equation assuming only a mildpointwise polynomial decay for the function f . Moreover, the solution we obtain has at mostexponential growth. We will prove the following.

Theorem 1. Let Mn be a complete Riemannian manifold of dimension n. Assume that M has pos-itive spectrum and Ricci curvature bounded below by a negative constant. Consider any boundedfunction f having decay

∣∣f (x)∣∣ � C

(1 + r(x))1+ε, (2)

where r(x) = d(p,x) and ε > 0. Then there exists a solution u of

�u = f.

Moreover, with respect to p this solution has at most exponential growth, i.e. there exist constantsA > 0 and B > 0 such that for any x ∈ M

∣∣u(x)∣∣ � AeBr(x).

200 O. Munteanu, N. Sesum / Advances in Mathematics 223 (2010) 198–219

Assuming that M has bounded geometry in a certain sense, we can improve our result and areable to show that the solution u has at most linear growth.

We then give several applications of Theorem 1. One of them is the existence of harmonicmaps which are homotopic to a given map and in that light we prove the following theorem.

Theorem 2. Assume that M is a complete Riemannian manifold with positive spectrum and Riccicurvature bounded below. Let N be a complete manifold with nonpositive sectional curvatureand h a smooth map from M to N. If with respect to a fixed point p ∈ M the tension field hasdecay at infinity

∥∥σ(h)∥∥(x) � C

(1 + r(x))1+ε,

then there exists a harmonic map u : M → N such that u is homotopic to h. Moreover, thehomotopic distance between u and h has at most exponential growth on M .

The second application is the existence of a Hermitian–Einstein metric on a holomorphicvector bundle over a complete Kähler manifold.

Theorem 3. Let M be a complete Kähler manifold with positive spectrum and Ricci curvaturebounded below. Assume that E is a holomorphic vector bundle of rank k over M with metric H0

such that

‖ΛFH0 − λI‖(x) � C

(1 + r(x))1+ε.

Then there exists a metric H on E such that ΛFH − λI = 0.

Assuming Lp conditions instead of pointwise decay, Theorem 2 was established in [5,10,14,18] and Theorem 3 was established in [19,20]. As we remarked above, our assumptions seemmore reasonable for manifolds with positive spectrum.

We also discuss the Ricci flow on complete Riemann surfaces. We consider a complete Rie-mann surface M with the scalar curvature S having a decay

∣∣S(x) + 1∣∣ = O

(r(x)−1−ε

), (3)

where ε > 0 and r(x) = d(p,x), with p ∈ M being a fixed point. This means in some sense ourmanifold is close to a hyperbolic surface at infinity. We will evolve such a metric by the Ricciflow equation,

∂

∂tg = −(S + 1)g, (4)

and study its long time existence and convergence as t → ∞.


Theorem 4. Let (M,g0) be simply connected Riemann surface with scalar curvature satisfying

−a2 � S � −b2 < 0,

and (3). Then the Ricci flow (4) starting at g0 exists forever and converges, as t → ∞, to acomplete metric of negative constant curvature.

The organization of the paper is as follows. In Section 2 we will give some preliminariesfor the proofs of our main results. In Section 3 we will prove Theorem 1. The proof includesvarious estimates on the growth of the Green’s function which are interesting in their own right.In Section 4 we will show how one can use Theorem 1 to prove Theorems 2 and 3 and will proveTheorem 4.

2. Preliminaries

In this section we will give some preliminaries about the Green’s function for a completemanifold M with positive spectrum and recall some terminology for harmonic maps betweentwo complete manifolds and Hermitian–Einstein metrics on holomorphic vector bundles overcomplete Kähler manifolds.

2.1. Green’s function

It is a standard result [23] that if the manifold has positive spectrum then it is nonparabolic,that is, there exists a positive symmetric Green’s function G on M . Moreover, we will always takeG(x,y) to be the minimal Green’s function, constructed using exhaustion of compact domains.

For any fixed x ∈ M and any 0 � α < β � ∞ denote by

Lx(α,β) = {y ∈ M: α < G(x,y) < β

},

lx(s) = {y ∈ M: G(x,y) = s

},

the level sets of G(x, ·). The dependence of Lx(α,β) and lx(s) on x will be quite important whenwe investigate the growth of a solution to the Poisson equation. We recall the following knownresults about the level sets of G:

(i) Since G is harmonic and with finite Dirichlet integral outside the pole,

∫lx (s)

|∇G|(x, y) dy

is a finite number independent of s.(ii) G has the following integral decay at infinity

∫Bx(R+1)\Bx(R)

G2(x, y) dy � C(x)e−2√

λ1(M)R, (5)

for any R > 2.


(iii) The co-area formula and (i) give for any δ > 0 and ε > 0 that

∫Lx(δε,ε)

G−1(x, y)|∇G|2(x, y) dy =ε∫

δε

1

tdt

∫lx (t0)

|∇G|(x, y) dy

=( ∫

lx (t0)

|∇G|(x, y) dy

)(− log δ).

For the proof of these results see [12].

2.2. Harmonic maps

For a map between two manifolds u : M → N define the energy density by

e(u) = gij (x)hαβ

(u(x)

)∂uα

∂xi

∂uβ

∂xj,

where ds2M = gij dxi dxj and ds2

N = hαβ dyα dyβ . Consider also σ(u) the tension field of amap u, i.e.

σ(u)(x) =(

�uα(x) + Γ αβδ

(u(x)

)∂uβ

∂xi

∂uδ

∂xjgij (x)

)∂

∂yα.

We will say the map u is harmonic if σ(u) = 0.

2.3. Hermitian–Einstein metric

Let us recall some notation. Let (E,H) be a holomorphic vector bundle with Hermitian met-ric H . Define the operator Λ as the contraction with the metric gij on M , i.e.

Λ(aij̄ dzi ∧ dz̄j

) = gij̄ aij̄ ,

for any (1,1) form a. The metric H is called Hermitian–Einstein if the corresponding curvatureFH satisfies

ΛFH = λI,

where λ is some constant.

3. The Poisson equation

The goal of this section is to prove Theorem 1. This result and the estimates we prove heremay have different applications for complete manifolds.

Let G(x,y) be the Green’s function as in Section 2. If we manage to show that for everyx ∈ M ,


∣∣∣∣∫M

G(x,y)f (y) dy

∣∣∣∣ < ∞,

then a function u(x) defined by the above integral solves (1). Moreover, we will show u(x) hasthe right exponential growth as stated in Theorem 1. In order to control the above integral weneed to control the Green’s function on M . That is the reason why we need to establish certaingrowth estimates for G(x, ·) first. Let us outline the main steps in the proof of Theorem 1 andthen give more details below.

(i) First we will show there are positive constants A and B such that for any x ∈ M and for anyy ∈ ∂Bx(1),

A−1e−Br(x) < G(x, y) < AeBr(x),

and that for any x ∈ M

∫lx (s)

|∇G|(x, y) dy = 1.

(ii) By maximum principle, for any x ∈ M we have Lx(AeBr(x),∞) ⊂ Bx(1). Using (i) weshow that

∫

Lx(AeBr(x),∞)

G(x, y) dy � CeBr(x).

A similar estimate appeared in [27].(iii) We will split the

∫M

G(x,y)f (y) dy into two parts,

∫

Lx(AeBr(x),∞)

G(x, y)f (y) dy and∫

Lx(0,AeBr(x))

G(x, y)f (y) dy.

To control the first one we will use the estimate in (ii) and to control the second one we willuse all the previous estimates and the decay for f (x) on M . The proof for existence of u

is based on a level set argument. We should point out that (i) and (ii) are used here only toprove the exponential growth of the u, the solution of (1).

Let us first prove some results about the behavior of Green’s function at infinity. These resultsare of independent interest. We will use the same symbol C for possibly different constants,which may depend on the fixed point p, the dimension of M , Ricci curvature lower bound andon the bottom of spectrum.

Lemma 1. There exist positive constants A and B such that for any x ∈ M and y ∈ ∂Bx(1) wehave the following estimates

A−1e−Br(x) < G(x, y) < AeBr(x).


Proof. Without loss of generality we can assume that x ∈ M\Bp(4).Consider σ (and τ ) the minimal geodesics joining p with x (and y).We divide the proof into two cases. In the first case, we assume that d(x, τ ) > 1

10 , whered(x, τ ) = mins d(x, τ (s)). Then G(x, ·) defines a harmonic function on Bz(1/20), for any z

on τ , and using Cheng–Yau’s gradient estimate [4] we conclude that there is a constant C suchthat

|∇G|(x, z) � CG(x, z),

for all z on τ . Integrating this along τ we find that

e−Cr(y)G(x,p) � G(x,y) � eCr(y)G(x,p).

However, using the gradient estimate again we find that

e−Cr(x)G(p,x0) � G(p,x) � eCr(x)G(p,x0),

for some x0 ∈ ∂Bp(1). This shows that there exist positive constants A and B , independent of x

and y, such that

A−1e−Br(x) < G(x, y) < AeBr(x).

In the second case, we assume that d(x, τ ) � 110 . Hence there exists a point z ∈ τ such that

d(x, z) � 110 . We claim that in this case d(y,σ ) > 1

10 . Suppose the contrary, that there exists apoint w ∈ σ such that d(y,w) � 1

10 . We then have:

d(p,x) � d(p, z) + d(z, x) � d(p, z) + 1

10,

d(p, y) � d(p,w) + d(w,y) � d(p,w) + 1

10.

Adding these two inequalities we get:

d(p,x) + d(p,y) � d(p,w) + d(p, z) + 1

5,

which implies that

d(w,x) + d(z, y) � 1

5.

In particular, this implies that

1 = d(x, y) � d(x,w) + d(w,y) � 1

5+ 1

10,

which gives a contradiction. We have thus shown that d(y,σ ) > 110 . Now we can conclude from

the argument in the first part of the proof that


A−1e−Br(x) < G(y, x) < AeBr(x).

The lemma is proved. �We want to point out that since M may contain cusp ends, in general this estimate in Lemma 1

is the best we may hope for.We prove a similar estimate for a level set integral for G. Recall that everywhere in this

paper G is the minimal symmetric Green’s function, hence

G(x,y) = limi→∞Gi(x, y),

where Gi is the Dirichlet Green’s function of a compact exhaustion {Ωi}i of M . The above limitis uniform on compact subsets of M .

Lemma 2. For any x ∈ M we have

∫lx (s)

|∇G|(x, y) dy = 1.

Proof. Consider Gi the Dirichlet Green’s function on Ωi , where {Ωi}i is a compact exhaustionof M . Then, by standard theory we know that for i sufficiently large such that Bx(1) ⊂ Ωi wehave

∫∂Bx(1)

∂Gi

∂r(x, y) dA(y) =

∫Bx(1)

�Gi(x, y) dy = −1.

Making i → ∞ it follows that

∫∂Bx(1)

∂G

∂r(x, y) dA(y) = −1.

Since G(x, ·) is harmonic on M\Bx(1) we get that indeed

∫lx (t)

|∇G|(x, y) dy = −∫

∂Bx(1)

∂G

∂r(x, y) dA(y) = 1.

This proves the lemma. �We are grateful to Mohan Ramachandran for showing us this proof of Lemma 2, which im-

proves and simplifies our original proof. We now prove another result about G(x,y) which isimportant for the behavior of the Green’s function near the pole.

Lemma 3. For A and B in Lemma 1 there exists a positive constant C such that for any x ∈ M

we have:


∫

Lx(AeBr(x),∞)

G(x, y) dy �∫

Bx(1)


Proof. We first notice that for any x ∈ M

Lx

(AeBr(x),∞) ⊂ Bx(1).

This is easy to see from Lemma 1, since by the maximum principle and the fact that G is theminimal Green’s function we have

supz∈M\Bx(1)

G(x, z) � supy∈∂Bx(1)

G(x, y) < AeBr(x).

Hence, our goal next is to justify that

∫Bx(1)


Claim 1. If the Ricci curvature is bounded below on a manifold M with positive spectrum and ifG1 is the Dirichlet Green’s function on a ball Bx(1), then there exists a constant C such that forany x ∈ M

∫Bx(1)

G1(x, y) dy < C. (6)

Proof. Notice that if H(z, y, t) is the Dirichlet heat kernel on Bx(1), we have

d

dt

∫Bx(1)

H 2(x, y, t) dy =∫

Bx(1)

2H�yH dy

= −2∫

Bx(1)

|∇yH |2(x, y, t) dy

� −2λ1(M)

∫Bx(1)

H 2(x, y, t) dy.

This proves that for t � 1 we have

∫Bx(1)

H 2(x, y, t) dy � e−2λ1(M)(t−1)

∫Bx(1)

H 2(x, y,1) dy.

Using that H(x,y, t) is less than that heat kernel on M and from the heat kernel estimates in [13]we get that there exists a constant C independent of x such that


supy∈Bx(1)

H(x, y,1) � C

Vol(Bx(1))1/2Vol(By(1))1/2.

The Bishop–Gromov volume comparison theorem implies that

1

Vol(By(1))� C

Vol(By(4))� C

Vol(Bx(1)),

therefore we obtain the following upper bound

supy∈Bx(1)

H(x, y,1) � C

Vol(Bx(1)).

Consequently, we get that

∫Bx(1)

H 2(x, y,1) dy � C

Vol(Bx(1)).

Now using the Cauchy–Schwarz inequality it follows that for t � 1

( ∫Bx(1)

H(x, y, t) dy

)2

� Vol(Bx(1)

) ∫Bx(1)

H 2(x, y, t) dy

� Ce−2λ1(M)(t−1).

The claim now follows using that

∫Bx(1)

G1(x, y) dy =∞∫

0

∫Bx(1)

H(x, y, t) dy dt.

Indeed, the estimates proved above show that

∞∫1

∫Bx(1)

H(x, y, t) dy dt � C,

whereas the contraction property

∫Bx(1)

H(x, y, t) dy � 1

will imply that∫ 1

0

∫Bx(1)

H(x, y, t) dy dt � 1. �The maximum principle now implies that for any y ∈ Bx(1)


G(x,y) � G1(x, y) + supz∈∂Bx(1)

G(x, z)

which combined with (6) and the Bishop–Gromov volume comparison proves that

∫Bx(1)

G(x, y) dy � C(

1 + supy∈∂Bx(1)

G(x, y)).

Lemma 3 now follows using Lemma 1. �After this preparation, we proceed to proving Theorem 1.

Proof of Theorem 1. Our goal is to show that there exist positive constants C and α such thatfor any x ∈ M ,

∣∣∣∣∫M

G(x,y)f (y) dy

∣∣∣∣ < Ceαr(x). (7)

Fix x ∈ M and as above denote by G(x,y) the minimal Green’s function on M with a poleat x.

The co-area formula (iii) and Lemma 2 yield for any δ > 0 and ε > 0,

∫Lx(δε,ε)

G−1(x, y)|∇G|2(x, y) dy =ε∫

δε

1

tdt

∫lx (t0)

|∇G|(x, y) dy

=( ∫

lx (t0)

|∇G|(x, y) dy

)(− log δ) = − log δ.

Notice that the integral decay of G from (ii) does not guarantee that (7) is true, as in general itis not enough to balance the exponential volume growth of M . The main idea of the proof of (7)is to use estimates on the level sets of G instead of estimates on geodesic balls. We write M asa disjoint union of sublevel sets of G, compute the integral of G on each such sublevel set byusing the co-area formula and λ1(M) > 0 and then estimate f on these sublevel sets, using thegiven decay (2) in order to show that (7) can be bounded above by a series that converges. Wefirst prove the following.

Claim 2. There exists a constant C such that for any 1 > δ > 0, any ε > 0 and for any x ∈ M

∣∣∣∣∫

Lx(δε,ε)

G(x, y)f (y) dy

∣∣∣∣ � C(− log δ + 1) supLx(δε,ε)

|f |. (8)

Proof. We follow a similar argument as in [17]. Let us choose the following cut off:

φ = χψ,


where we define

χ(y) =

⎧⎪⎪⎨⎪⎪⎩

(log 2)−1(logG(x,y) − log( 12δε)) on Lx(

12δε, δε),

1 on Lx(δε, ε),

(log 2)−1(log 2ε − logG(x,y)) on Lx(ε,2ε),

0 otherwise

and

ψ(y) =⎧⎨⎩

1 on Bx(R),

R + 1 − d(x, y) on Bx(R + 1)\Bx(R),

0 on M\Bx(R + 1).

Then we have:

λ1(M)

∫Lx(δε,ε)∩Bx(R)

G(x, y) dy

� λ1(M)

∫M

G(x,y)φ2(y) dy �∫M

∣∣∇(G

12 φ

)∣∣2(x, y) dy

� 1

2

∫

Lx( 12 δε,2ε)

G−1(x, y)|∇G|2(x, y) dy + 2∫M

G(x,y)|∇φ|2(y) dy,

where we have used the Poincaré inequality on M . We now compute each term above. Accordingto the co-area formula (iii) and Lemma 2, we know that

∫

Lx( 12 δε,2ε)

G−1(x, y)|∇G|2(x, y) dy � (− log δ + C). (9)

On the other hand, we have

∫M

G(x,y)|∇φ|2 � 2∫M

G(x,y)|∇χ |2ψ2 + 2∫M

G(x,y)|∇ψ |2χ2

� 4∫

Lx( 12 δε,2ε)

G−1(x, y)|∇G|2(x, y) dy

+ 2∫

Bx(R+1)\Bx(R)

G(x, y)χ2(y) dy

� C(− log δ + 1) + C(x)

δεe−2

√λ1(M)R.

In the last step we have used (9) and that since we are on the support of χ , in particular G(x,y) �1δε. Then we also have used (5) to estimate
2


∫Bx(R+1)\Bx(R)

G(x, y)χ2 dy � 2

δε

∫Bx(R+1)\Bx(R)

G2(x, y) dy

� C(x)

δεe−2

√λ1(M)R.

We have thus proved that

∫Lx(δε,ε)∩Bx(R)

G(x, y) dy � C(− log δ + 1) + C(x)

δεe−2

√λ1(M)R.

Making now R → ∞ we get that

∫Lx(δε,ε)

G(x, y) dy � C(− log δ + 1).

Finally by

∣∣∣∣∫

Lx(δε,ε)

G(x, y)f (y) dy

∣∣∣∣ � supLx(δε,ε)

|f |∫

Lx(δε,ε)

G(x, y) dy

we conclude the proof of the claim. �Furthermore, we have

∣∣∣∣∫M

G(x,y)f (y) dy

∣∣∣∣ �∣∣∣∣

∫

Lx(AeBr(x),∞)

G(x, y)f (y) dy

∣∣∣∣

+∣∣∣∣

∫

Lx(0,AeBr(x))

G(x, y)f (y) dy

∣∣∣∣. (10)

We will estimate each of these integrals in (10). We start with the first integral.Since f is bounded on M , Lemma 3 implies that

∣∣∣∣∫

Lx(AeBr(x),∞)

G(x, y)f (y) dy

∣∣∣∣ � CeBr(x). (11)

We want to estimate the second term in (10).For any z ∈ M\Bx(1) consider γ the minimal geodesic joining x and z. Let z0 ∈ ∂Bx(1) be

the intersection of this geodesic with ∂Bx(1). Then we have, using the gradient estimate and thelower bound of G from Lemma 1, that

G(x, z) � G(x, z0)e−Cd(z0,z) � A−1e−Br(x)e−C0d(x,z).


We prefer to write this in the following equivalent way:

G(x, z) � e− logA−Br(x)−C0d(x,z), (12)

for all z ∈ M\Bx(1). The constants A,B and C0 do not depend on x or z. For these fixed con-stants let

m0 = 1 + max{2(B + C0)r(x) + 2 logA,Br(x) + logA

}.

This choice of constant m0 in particular implies that

Lx

(0, e−m0

) ⊂ Lx

(0,A−1e−Br(x)

) ⊂ M\Bx(1),

by Lemma 1 and the maximum principle.Let us write

∣∣∣∣∫

Lx(0,AeBr(x))

G(x, y)f (y) dy

∣∣∣∣ �∣∣∣∣

∫

Lx(0,e−m0 )

G(x, y)f (y) dy

∣∣∣∣

+∣∣∣∣

∫

Lx(e−m0 ,AeBr(x))

G(x, y)f (y) dy

∣∣∣∣. (13)

We now estimate | ∫Lx(0,e−m0 )

G(x, y)f (y) dy| from above. First, note that from (8) it followsthat:

∣∣∣∣∫

Lx(0,e−m0 )

G(x, y)f (y) dy

∣∣∣∣ �∑

m�m0

∣∣∣∣∫

Lx(e−(m+1),e−m)

G(x, y)f (y) dy

∣∣∣∣

� C∑

m�m0

supLx(e−(m+1),e−m)

|f |. (14)

Claim 3. The following series can be bounded from above by a constant C independent of x:

∑m�m0


|f | � C < ∞. (15)

Proof. The choice of constant m0 implies that Lx(0, e−m0) ⊂ M\Bx(1), therefore using (12) weget that

G(x, z) � e− logA−Br(x)−C0d(x,z),

for all z ∈ Lx(0, e−m0). Consequently, on Lx(e−(m+1), e−m) we have

e−m � G(x, z) � e− logA−Br(x)−C0d(x,z).


This implies that

d(x, z) � m − logA − Br(x)

C0.

Moreover, we claim that

d(p, z) >1

2C0m,

for all z ∈ Lx(e−(m+1), e−m). To see this note that

d(p, z) � d(x, z) − d(p,x)

� m − logA − Br(x)

C0− r(x)

>m

2C0

where the last inequality holds if and only if m > 2(B + C0)r(x) + 2 logA, which is clearlysatisfied by our choice of m0.

This argument shows that

Lx

(e−(m+1), e−m

) ⊂ M\Bp

(1

2C0m

),

which we now use to estimate

∑m�m0


|f | �∑

m�m0

supM\Bp( 1

2C0m)

|f |

�∑

m�m0

C

(1 + 12C0

m)1+ε� C.

This completes the proof of (15). �By Claim 3 and estimate (14) we have

∣∣∣∣∫

Lx(0,e−m0 )

G(x, y)f (y) dy

∣∣∣∣ � C. (16)

The other term in (13) we estimate using (8) and that f is bounded on M :

∣∣∣∣∫

Lx(e−m0 ,AeBr(x))

G(x, y)f (y) dy

∣∣∣∣ � C log(AeBr(x)+m0

)� C

(r(x) + 1

).

Therefore, by (13) and (16) we conclude that


∣∣∣∣∫

Lx(0,AeBr(x))

G(x, y)f (y) dy

∣∣∣∣ � C(r(x) + 1

). (17)

Combining (10), (11) and (17) yields

∣∣∣∣∫M

G(x,y)f (y) dy

∣∣∣∣ < Ceαr(x).

This concludes the proof of (7) and hence of Theorem 1. �Corollary 1. Let M be a complete Riemannian manifold with positive spectrum and Ricci curva-ture bounded from below. Assume moreover that there exists a constant a > 0 such that for anyx ∈ M and for any y ∈ ∂Bx(1)

G(x, y) < a(r(x) + 1

).

Consider any function f with decay at infinity as in Theorem 1. Then the Poisson equation�u = f admits a solution u of at most linear growth, i.e. there exists b > 0 such that for anyx ∈ M

∣∣u(x)∣∣ � b

(r(x) + 1

).

Proof. The argument in Lemma 3 implies that there exists a constant C such that for any x ∈ M ,

∣∣∣∣∫

Lx(a(r(x)+1),∞)

G(x, y)f (y) dy

∣∣∣∣ � C(r(x) + 1

).

We now justify, following the proof of (17), that

∣∣∣∣∫

Lx(0,a(r(x)+1))

G(x, y)f (y) dy

∣∣∣∣ � C(r(x) + 1

). (18)

We have (using Lemma 1) that for any x ∈ M and any y ∈ ∂Bx(1)

A−1e−Br(x) < G(x, y) < a(r(x) + 1

).

The argument in Theorem 1 implies that

∣∣∣∣∫

Lx(0,a(r(x)+1))

G(x, y)f (y) dy

∣∣∣∣ � C∑

m�m0

supLx(e−m−1,e−m)

|f |

+∣∣∣∣

∫−m0

G(x,y)f (y) dy

∣∣∣∣,
Lx(e ,a(r(x)+1))


for m0 the same as in Theorem 1. We can use the same argument as in Theorem 1 to establish(12) and (15), therefore

∑m�m0

supLx(e−m−1,e−m)

|f | � C.

Finally, from (8) we get that

∣∣∣∣∫

Lx(e−m0 ,a(r(x)+1))

G(x, y)f (y) dy

∣∣∣∣ � C log(a(r(x) + 1

)em0

)� C

(r(x) + 1

),

and this proves (18) and hence the result. �To state our last result of this section, we now assume that M is a complete Riemannian

manifold with positive spectrum, Ricci bounded from below and that M has a bounded geometryof the following type. Assume that there exists r0 > 0 with the property that for any ball Bx(r0)

there exists a chart χx : Bx(r0) → Rn such that

c−10 d(y, z) �

∥∥χx(y) − χx(z)∥∥ � c0d(y, z),

for any x ∈ M and for any y, z ∈ Bx(r0).

Corollary 2. Assume that M is a complete Riemannian manifold with positive spectrum, Riccicurvature bounded from below and bounded geometry as above. Then for any f that has decay atinfinity as in Theorem 1, the Poisson equation �u = f has a solution u of at most linear growth.

The proof of this Corollary follows from Ancona’s work in [1]. Indeed, Proposition 7 in [1]states that G(x,y) will have the property required by Corollary 1, hence the result.

4. Applications to harmonic maps and Hermitian–Einstein metrics

In this section we first use the solution of the Poisson equation to investigate the existence ofharmonic maps which are homotopic to a given map. Then we use a similar argument to studythe existence of a Hermitian–Einstein metric on a holomorphic vector bundle over a completemanifold.

Results like Theorem 2 were previously proved by Li and Tam [10] and Ding and Wang [5],assuming that e(h) ∈ Lp(M) for p finite. In addition to the existence result, they were also ableto prove that the homotopy distance between u and h is in Lp(M). The proof of Li and Tam wasbased on a heat equation approach, using ideas of Eells and Sampson from the compact setting.Here we use an elliptic approach, following the arguments in [14,18,19]. We only sketch theargument here and point out where the Poisson equation is being used; for complete details werefer to the cited papers.

Proof of Theorem 2. Let Ωi be a compact exhaustion of M with at least Lipschitz boundary.It is known that by a result of Hamilton [7] there exists a sequence ui that solves the Dirichlet

homotopy problem


σ(ui) = 0,

ui |∂Ωi= h,

ui ∼ h rel ∂Ωi.

In order to show convergence of (ui) to a harmonic map u : M → N , it is enough to prove localboundedness of the energy density functions e(ui). To achieve this consider ρi the homotopicdistance between ui and h and ρij the homotopic distance between ui and uj . It can be provedthat a uniform bound on ρij implies a uniform bound on the energy density e(ui), see [18]. Thisargument only uses local geometry and it is true in our situation, too. Therefore if we can provethat the sequence ρij is uniformly bounded on compact sets, then the convergence of u followsfrom this argument. We now prove a uniform bound for ρi , which by ρij � ρi + ρj implies auniform bound for ρij . This is where we use the solution to the Poisson equation from Theorem 1.Recall that ρi satisfy a fundamental differential inequality [22]

�ρi � −∥∥σ(h)∥∥,

where ‖σ(h)‖ is the norm of the tensor field. From the hypothesis we know that ‖σ(h)‖ hasthe right decay so we may apply Theorem 1. This gives a positive function v that solves �v =−‖σ(h)‖. We now use the maximum principle to see that

ρi � v.

Indeed, v − ρi is superharmonic on Ωi and it is positive on ∂Ωi . This proves that ρi is uni-formly bounded on a fixed compact set K ⊂ M , therefore the energy density e(ui) is uniformlybounded on K . This proves the existence of u. Since by Theorem 1 the function v has at mostthe exponential growth, it follows that the homotopic distance between u and h grows at mostexponentially. �

We now discuss another application of Theorem 1, which concerns the existence ofHermitian–Einstein metrics on holomorphic vector bundles over a complete Kähler manifold,stated in Theorem 3.

The existence of a Hermitian–Einstein metric on a compact manifold is related to stabilityof the vector bundle, as it is well known from the work of Donaldson and Uhlenbeck–Yau. Oncomplete manifolds Ni [19] and Ni and Ren [20] have shown that if there exists a Hermitianmetric H0 such that ‖ΛFH0 − λI‖ ∈ Lp(M) for some p � 1 finite and M has positive spectrumthen there exists a metric H such that ΛFH = λI . As in the case of harmonic maps, we canmodify their argument to obtain Theorem 3.

Proof of Theorem 3. The argument here is similar to the argument for harmonic maps. We useagain our solution from Theorem 1 and the maximum principle to prove certain C0 estimates.We discuss here the special case λ = 0, however the general case follows the same. By a resultof Donaldson [6] there exists a hermitian metric Hi on Ωi such that

ΛFHi= 0 on Ωi,

Hi |∂Ω = H0.
i


The goal is to prove that we can pass to a limit and obtain a solution on M and for this weestablish a priori estimates. Recall the following distance functions introduced by Donaldson

τi(x) = τ(Hi,H0) = tr(HiH

−10

),

σi(x) = σ(Hi,H0) = tr(HiH

−10

) + tr(H0H

−1i

) − 2 rank(E).

In order to prove that Hi has a subsequence which converges uniformly on compact subsets ofM we need to establish C0 and C1 estimates. The C0 estimates are based on the solution tothe Poisson equation found in Theorem 1. Recall that we have a Bochner type inequality for thedistance τi , see [19,24]. Let fi = log τi − logk, where k is the rank of E. Then

�fi � −‖ΛFH0‖ on Ωi,

fi = 0 on ∂Ωi.

Using the maximum principle it is easy to see that

fi � v,

where v is the solution of

�v = −‖ΛFH0‖.Such a solution exists and has at most exponential growth since the norm of ΛFH0 has the

decay as in Theorem 1. This shows that τi � kev and therefore we get a bound for σi

σi � 2kev − 2k.

This establishes the C0 estimates and now the C1 estimates follow the same as in [19]. Since theargument for these C1 estimates only uses local geometry it is valid in our situation as well. Thisconcludes the proof. �4.1. The Ricci flow

In order to prove Theorem 4 we want to find a bounded solution to the Poisson equation

�u = S + 1,

at time t = 0, where � and S are the Laplacian and scalar curvature taken with respect to met-ric g0. Then, Theorem 4 will follow as an immediate consequence of a result in [3]. We also wantto point out that the argument in [8] (also see [9]) implies that the existence of a solution u withbounded gradient would also suffice to show that Theorem 4 holds true. However, is not clear tous how to improve our results from Section 3 to deduce that the solution u is in fact bounded orwith bounded gradient.

We will assume below that M is a Cartan Hadamard manifold with Ricci curvature pinchedby two negative constants. It is well known that such manifolds have positive spectrum, thereforeour results in Section 3 apply here. In fact, a standard argument using the Laplace comparison


theorem will give that the solution to the Poisson equation is moreover bounded. The followingresult holds in arbitrary dimension.

Proposition 1. Let (M,g0) be a Cartan Hadamard manifold of dimension n. Assume that itsRicci curvature satisfies

−a2 � RicM � −b2 < 0.

Then for any f ∈ C∞(M) having a decay

∣∣f (x)∣∣ � C

(1 + r(x))1+ε

where ε > 0 and r(x) is a distance from x to a fixed point p, there exists a unique solution to�u = f having decay

∣∣u(x)∣∣ � C

(1 + r(x))ε.

Proof. Consider the sequence ui : Bp(Ri) → R, for Ri → ∞, such that

�ui = f, on Bp(Ri),

ui = 0, on ∂Bp(Ri).

Our goal is to show that we have uniform estimates on ui so that passing to a limit as i → ∞yields to a global solution to the Poisson equation �u = f with a desired decay at infinity.Observe that by Laplacian comparison theorem (see [26, Theorem 2.15])

�r � b coth(br) � b,

where r(·) = d(·,p). For α > 0 to be determined later we have

�(r + α)−ε = −ε(r + α)−ε−1�r + ε(ε + 1)(r + α)−ε−2

� −ε(r + α)−ε−1(

b − ε + 1

r + α

).

By choosing α = max{1,2(ε+1)

b} we see that

�(r + α)−ε � −1

2εb(r + α)−ε−1. (19)

For the same constant C as in the decay of f define

A = 2C

bεα1+ε.

Consider


vi = A(r + α)−ε − ui.

Then by (19), the decay of f and the value of A we get

�vi � −1

2Aεb(r + α)−ε−1 + C(1 + r)−ε−1 � 0.

By the maximum principle we get that vi > 0 on Bp(Ri) and therefore

ui � A(r + α)−ε.

The same arguments applied to −ui imply

|ui | � A

(r + α)ε, on Bp(Ri).

This shows ui is uniformly bounded hence letting i → ∞ we conclude the proof. �Proof of Theorem 4. By Proposition 1 we can find a bounded potential solving

�u = S + 1.

By Theorem 1.1 in [3] it immediately follows the Ricci flow has a long time existence on M andit converges, as t → ∞, uniformly on compact subsets, to a Kähler–Einstein metric with constantnegative curvature. �Acknowledgments

The authors would like to thank Mohan Ramachandran for stimulating conversations and forproviding the reference [1].

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the poisson equation on complete manifolds with positive spectrum and applications

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