A Note on the Existence of Positive Green's Function

Journal of Functional AnalysisFU3235 journal of functional analysis 156, 199207 (1998) article no. FU973235

A Note on the Existence of Positive Green's Function Chiung-Jue Anna Sung* Department of Mathematics, National Chung Cheng University, Chia-Yi, Taiwan 62117 E-mail: cjsungmath.ccu.edu.tw Received November 5, 1997; revised December 9, 1997

It is an important problem to determine when a complete noncompact Riemannian manifold admits a positive Green's function. In this regard, one tries to seek geometric assumptions which are stable with respect to uniform perturbations of the metric. In this note, we obtained some results in this direction, generalizing some earlier results of Li and Tam.  1998 Academic Press

1. INTRODUCTION The purpose of this note is to establish the existence of positive Green's function on a class of complete noncompact Riemannian manifolds satisfying some geometric assumptions that are invariant under quasi-isometries. Green's function, as the fundamental solution to the Poisson equation, is obviously an important object, as the problem regarding its existence and estimates has been extensively studied. In 1955, Malgrange [M] proved the existence of a symmetric Green's function on any complete Riemannian manifold. Later, Li and Tam [LT2] have given an alternative argument to this result. Their proof showed that a symmetric Green's function can be constructed via compact exhaustion. This approach turned out to be important and very useful in understanding the problem when a Riemannian manifold is nonparabolic. Namely, the manifold has a positive Green's function. There are necessary conditions for the existence of a positive Green's function. In 1975, Cheng and Yau [CY] obtained first result in this direction, which involves only the volume growth of the complete manifold. The result was later improved by Varopoulos in [V4], which states that if a complete manifold M has a positive Green's function, then

|



0

t dt<. V p(t)

(0.1)

* The author was partially supported by NSC of Taiwan.

199 0022-123698 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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200

CHIUNG-JUE ANNA SUNG

Here and in the following, we shall let V p(t) denote the volume of geodesic ball of radius t with center at p in M. However, the condition (0.1) is far from sufficient as pointed out in [V4]. The sharp necessary condition of Varopoulos was also later proved in [LT3] and [G1] by using different arguments. The first major result for the sufficiency was due to Varopoulos [V1] and Li and Yau [LY]. Using the estimates for the heat kernel, Li and Yau proved that if M has non-negative Ricci curvature everywhere, and if (0.1) is satisfied, the M will have a positive Green's function. Furthermore, the minimal positive Green's function of M satisfies

C &1

t dtG(x, y)C r(x, y) V x (t)

|



t dt r(x, y) V x (t)

|



(0.2)

for all x{ y in M, where constant C>0 depending only on the dimension of M. Later it was proved, in [LT1], that the condition (0.1) is also sufficient for the existence of a positive Green's function on a complete manifold with non-negative sectional curvature outside a compact subset. Moreover, the estimate (0.2) is still valid at each large end, where the constant C now depends on the manifold and also the point x. The result was later generalized to the asymptotically non-negatively curved manifolds by Kasue [K2]. The results in [LT1] and [K2] in one sense are generalizations of the results in [V1] and [LY], because there is no curvature restriction on a compact set. In another sense the results are more restrictive since the assumption is on the sectional curvature rather than on the Ricci curvature. Li and Tam [LT4] have succeeded in extending the result to manifolds satisfying suitable Ricci curvature assumptions. Among other things, they showed that if the Ricci curvature of the manifold M satisfies Ric M(x)&(n&1) K(1+r(x)) &2, and if, in addition, the manifold satisfies a volume comparison condition (VC) (see Definition 4), then condition (0.1) is again sufficient to guarantee the existence of a positive Green's function. While all these results are very deep and interesting, it should be noted that nonparabolicity of a manifold is an invariant property under quasi-isometries (see [LSW]). Thus, it is natural to seek sufficient conditions for nonparabolicity which are invariant under quasiisometries. This paper is one step in this direction. Our main result says that if the first Neumann eigenvalue of B x (r) satisfies * 1(B x (r))C 1 r 2 and V x (2r)c 2 V x (r) for all balls B x (r) with 2rd(x, p), where p # M is a fixed point and B x (r) is the geodesic ball of radius r and centered at x, then M is nonparabolic provided that the volume growth of M along each geodesic ray from point p is large. We refer to Section 2 for a precise statement. In particular, our result generalizes all the proceding results. It is also clear that our assumptions are invariant under quasi-isometries.

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201

EXISTENCE OF POSITIVE GREEN'S FUNCTION

2. EXISTENCE OF GREEN'S FUNCTION AND ITS ESTIMATES In this section, we will prove our main result concerning the existence of positive Green's function. Our argument utilizes the ideas originated in [LT4]. At the end, we will mention some corollaries. Let us start by recalling the definition of Green's function. Denote by 2 the Laplace Beltrami operator on an n-dimensional Riemannian manifold (M n, g) with metric g. Then, under local coordinates (x 1 , x 2 , ..., x n ), 2g =

1 -G

n

: i, j=1





g -G , x \ x + ij

i

j

where G=det( g ij ) and ( g ij )=(g ij ) &1. Definition 1. Let (M n, g) be a Riemannian manifold with or without boundary and 2 the associated LaplaceBeltrami operator. A function G(x, y) is called a Green's function of (M, g) if it satisfies (1)

G(x, y)=G( y, x);

(2)

for any fixed y # M, G(x, y) # C 2(M "[x]) as a function of x;

(3) 2 y G(x, y)=&$ y(x) as distributions, where $ y(x) is the usual delta function which picks the function value at point y when it applies to any continuous function with compact support on M. If M is not empty, then G(x, y) is said to satisfy the Dirichlet boundary conditions if G(x, y)=0 for x # M and the Neumann boundary conditions if G(x, y) &

}

, M=0

where & is the normal vector of M. We will need the following lemma in proving our result. Lemma 2. Let M n be a complete noncompact manifold and p # M a fixed point. Suppose that there are constants c 1 , c 2 >0 such that the first Neumann eigenvalue of B x (r) satisfies * 1(B x (r))c 1 r 2 and V x (2r)c 2 V x (r) for all balls B x (r) with 2rd(x, p). Let f be a harmonic function on M "B p(r 0 ). Then for all x # M "B p(r 0 ) and rd(x, p)&r 0 , osc Bx (r4) f 

cr - V x (r2)

\|

Bx (r2)

|{f | 2 ( y) dy

+

12

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.

202 Proof.

CHIUNG-JUE ANNA SUNG

Let f be a harmonic function on M"B p(r 0 ). For any constant c, 2( f &c) 2 =2( f &c) 2( f &c)+2 |{( f &c)| 2 =2 |{f | 2.

Therefore, the function ( f &c) 2 is subharmonic on M "B p(r 0 ). From our assumptions and the result of Grigor'yan [G2] and Saloff-Coste [SC], we may apply the mean value inequality to the function ( f &c) 2 on B x (r) and conclude that for some constant c 3 >0, sup ( f &c) 2  Bx (r4)

c3 V x (r2)

|

( f &c) 2.

(1.1)

|{f | 2,

(1.2)

Bx (r2)

On the other hand, since * 1(B x (r2))4c 1 r 2,

|

|

( f &c 0 ) 2 c 4 r 2

Bx (r2)

Bx (r2)

where c 0 = Bx (r2) fV x (r2). Combining (1.1) and (1.2) and setting c=c 0 in (1.1), we get sup ( f &c 0 ) 2  Bx (r4)

c5 r2 V x (r2)

|

|{f | 2.

(1.3)

Bx (r2)

Rewrite (1.3) into sup | f &c 0 |  Bx (r4)

cr - V x (r2)

\|

|{f | 2 Bx (r2)

+

12

.

(1.4)

For y 1 and y 2 in B x (r2), Eq. (1.4) implies | f ( y 1 )& f ( y 2 )|  | f ( y 1 )&c 0 | + | f ( y 2 )&c 0 | 

cr

\|

- V x (r2)

|{f | 2 Bx (r2)

+

12

.

Hence, osc Bx (r4) f 

cr - V x (r2)

\|

Bx (r2)

|{f | 2

+

12

,

and the lemma is proved. Before we state and prove our main result, let us first recall the construction of a symmetric Green's function by compact exhaustion on a general

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EXISTENCE OF POSITIVE GREEN'S FUNCTION

203

complete Riemannian manifold, which was carried out by Li and Tam in [LT2]. For a fixed point p # M, consider the Green's function G R(x, y) defined on the geodesic ball B p(R)/M satisfying Dirichlet boundary conditions. Then it was shown in [LT2] that there exists R i A  and constant c i such that G(x, y)= lim (G i (x, y)&c i ) i

exists for all x{ y in M and the convergence is uniform on compact subset of M, where we wrote G i (x, y) instead of G Ri (x, y) to simplify the notations. It is clear that such G(x, y) is a symmetric Green's function of M. Note that in general such G(x, y) is not necessarily unique. However, if M has a positive Green's function, then one can take c i =0 and the resulting G(x, y) will be the minimal positive Green's function of M. In the following, we shall let G( p, x) be the Green's function constructed from the compact exhaustion as described and give an estimate of G( p, x) from below. In particular, we obtain sufficient conditions to guarantee the existence of a positive Green's function for a class of manifolds. We point out that our assumptions do not involve the curvature of a manifold under consideration and are invariant with respect to quasi-isometries. Theorem 3. Let M n be a complete noncompact manifold and p # M a fixed point. Suppose that there are constants c 1 , c 2 >0 such that the first Neumann eigenvalue of B x (r) satisfies * 1(B x (r))c 1 r 2 and V x (2r) c 2 V x (r) for all balls B x (r) with 2rd(x, p). Let G( p, x) be the Green's function constructed by compact exhaustion. Then there exist a constant c>0 such that G( p, x)&c&c sup ;

|

d(x, p) 1

t dt, V ;(t)(t2)

where the supremum is taken over all the minimal geodesics ; from p to B p(r). Proof. By the maximum principle, it suffices to establish the estimate for x # M with d(x, p)=2 k for all integers k. For r>2, choose k1 such that 2 k 0, let s(r)=sup Bp(r) g and i(r)= inf Bp(r) g. Further, we choose y # B p(2 k ) such that g( y)= sup g=s(2 k ). Bp(2k )

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204

CHIUNG-JUE ANNA SUNG

Let ;(t) be a minimal geodesic from p to y parameterized by arc length. Let y i =r(2 i ), i1, so that y= y k . Applying the lemma to the function g, we have | g( y i+1 )& g( y i )|  

c2 i - V yi+1(2i } 32) c2 i - V yi+1(2i } 32)

\| \|

|{g| 2

By

i+1

(2i } 32)

+

12

|{g| 2

Bp(3 } 2k&1 )"Bp(1)

+

12

.

Summing over i=1, ..., k, we get

_\| c | \ V

|{g| 2

g( y)& g(r(2))c

Bp(3 } 2k&1 )"Bp(1)

2k

1

t

dt

;(t)(t2)

+

12

+

12 k&1

: i=1

2i - V yi+1 (2 i } 32)

\|

|{g| 2

Bp(3 } 2k&1 )"Bp(1)

+

&

12

,

where we have used the inequality 2i - V yi+1 (2 i } 32)

\

 c

|

2i+1 2i

t V ;(t)(t2)

+

12

,

which can be easily verified. From the construction of G(x, y) and the result in [LT3], we have

|

|{g| 2 4(s(3 } 2 k&1 )&i(1)).

Bp(3 } 2k&1 )"Bp(1)

Therefore, s(2 k )& g(r(2))c(s(3 } 2 k&1 )&i(1))

|

2k 1

t dt. V ;(t)(t2)

Now our assumptions imply that a version of Harnark inequality holds on B x ((78) 2 k ) for r(x, p)=2 k as shown in [G2] and [SC]. In particular, we conclude that sup Bx ((78) 2k )

( g&i(1))c

inf

( g&i(1)).

Bx ((78) 2k )

Hence, s(3 } 2 k&1 )&i(1)c(s(2 k )&i(1)).

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EXISTENCE OF POSITIVE GREEN'S FUNCTION

205

Thus, s(2 k )&i(1)s(2 k )& g(r(2))+ g(r(2))&i(1) c(s(2 k )&i(1)) 12 c

|

2k 1

\|

2k 1

t dt V ;(t)(t2)

+

12

+ g(r(2))&i(1)

t dt+c(s(2)&i(1)). V ;(t)(t2)

In conclusion, we obtain sup [&G( p, x)]c+c sup Bp(2k )

;

|

2k 1

t dt, V ;(t)(t2)

where ; is a minimal geodesic from p to B p(2 k ). Now the maximum principle implies that G( p, x)&c&c sup ;

|

d( p, x) 1

t dt V ;(t)(t2)

for all x # M. This proves the theorem. To conclude, we shall state several corollaries to our theorem. First, let us recall the definition of the volume comparison property for a complete manifold, which was introduced by Li and Tam in [LT4]. Definition 4. A complete noncompact manifold M satisfies a volume comparison condition (VC ), if there exists a constant `>0 such that for all x # B p(r), V p(r)`V x (r2) where V x (r) denotes the volume of the geodesic ball B x (r) of radius r with center at x. It should be clear that for a manifold with (VC) property, sup ;

|

 1

t dt< V ;(t)(t2)

where the supremum is taken over all the geodesic rays ; from p if and only if

|



1

t dt<. V ;(t)(t2)

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206

CHIUNG-JUE ANNA SUNG

Corollary 5. Let M be as in Theorem 3. If sup ;

|

 1

t dt<, V ;(t)(t2)

where the supremum is taken over all the geodesic rays of ; emanating from p, then M has a positive Green's function. Furthermore, if M has (VC ) property, then M is nonparabolic if and only if

|



1

t dt<. V p(t)

The following corollary generalizes the result of Li and Tam mentioned in the first section. Corollary 6. Suppose that M has asymptotically nonnegative Ricci curvature in the sense that Ricci(x)&c(1+d(x, p)) &2. Suppose also sup ;

|

 1

t dt<, V ;(t)(t2)

where the supremum is taken over all the geodesic rays of ; emanating from p. Then M has a positive Green's function. Furthermore, if M has (VC ) property, then M is nonparabolic if and only if

|



1

t dt<. V p(t)

Proof. The assumption on the Ricci curvature implies that M satisfies the assumptions in Theorem 3 (see [SC]).

ACKNOWLEDGMENTS I thank Professor Peter Li and Jiaping Wang for their interest in this work.

REFERENCES [CY] [G1]

S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333354. A. A. Grigor'yan, On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds (in Russian), Math. Sbornik 128(3) (1985), 354363. [English Translation in Math. USSR Sbornik 56 (1992), 349358]

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EXISTENCE OF POSITIVE GREEN'S FUNCTION [G2]

[K1] [K2] [LSW]

[LT1] [LT2] [LT3] [LT4] [LY] [M] [S] [SC] [V1] [V2] [V3] [V4]



A. A. Grigor'yan, The heat equation on noncompact Riemannian manifolds (in Russian), Math. Sbornik 182(1) (1991), 5587 [English Translation in Math. USSR Sbornik 72 (1) (1992), 4777]. A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. Ecole Norm. Sup. 22 (1988), 593622. A. Kasue, ``Lecture notes in Math.,'' Vol. 1339, pp. 158181, Springer-Verlag, Berlin New York, 1988. W. Lettman, G. Stampcchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa 17 (1960), 4377. P. Li and L. F. Tam, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. Math. 125 (1987), 171207. P. Li and L. F. Tam, Symmetric Green's functions on complete manifolds, Amer. J. Math. 109 (1987), 11291154. P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1993), 359383. P. Li and L. F. Tam, Green's functions, harmonic functions and volume comparison, J. Differential Geom. 41 (1995), 277318. P. Li and S. T. Tau, On the parabolic kernel of the Schrodinger operator, Acta. Math. 156 (1986), 153201. M. Malgrange, Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution, Ann. Inst. Fourier 6 (1955), 271355. C. J. Sung, Harmonic Functions under quasi-isometry, J. Geom. Anal. (to appear). L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417450. N. Varopoulos, The Poisson kernel on positively curved manifolds, J. Funct. Anal. 44 (1981), 359380. N. Varopoulos, Green's functions on positively curved manifolds, J. Funct. Anal. 45 (1982), 109118. N. Varopoulos, Green's functions on positively curved manifolds, II, J. Funct. Anal. 49 (1982), 170176. N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, in ``Proceedings, Conference on Harmonic Analysis in Honor of Antoni Zygmund,'' Wadsworth Math. Ser., Vols. I, II, pp. 821837, Wadsworth, Belmont, CA, 1983.















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