Harmonic Functions on Complete Riemannian Manifolds

Transcription

1 * Vol. I?? Harmonic Functions on Complete Riemannian Manifolds Peter Li Abstract We present a brief description of certain aspects of the theory of harmonic functions on a complete Riemannian manifold. The emphasis is on some of the developed techniques in the subject and also geometric applications that followed Mathematics Subject Classification: 58-00, 58J00. Keywords and Phrases: Harmonic functions, Complete manifolds, Green s functions, Harmonic maps. Table of Contents 0 Introduction 1 Gradient Estimates 2 Green s Function and Parabolicity 3 Heat Kernel Estimates and Mean Value Inequality 4 Harmonic Functions and Ends 5 Stability of Minimal Hypersurfaces 6 Polynomial Growth Harmonic Functions 7 Massive Sets and the Structure of Harmonic Maps 8 L q Harmonic Functions References 0 Introduction The purpose of this note is to give a rough overview of the subject of harmonic functions on complete Riemannian manifolds. Since the seminal work of Yau [Y1] in 1985, where he proved a Liouville theorem for positive harmonic functions on a complete manifold with nonnegative Ricci curvature, the issue of understanding various spaces of harmonic functions has been one of the central questions in geometric analysis. During the last 20 years, there had been many significant Research partially supported by NSF grant DMS Department of Mathematics, University of California, Irvine CA , USA. pli@math.uci.edu

2 190 Peter Li discoveries, but more importantly, the techniques developed in these theories are extremely useful when applied to other problems in geometic analysis. The point of view taken in these notes is to describe some of the tools and techniques developed in the context of harmonic functions. The primary goal is to present the subject to the extend that it is instructional and provide a glimpse of the issues involved. As a result, this will not be, and it is not meant to be, a comprehensive treatment to the theory of harmonic functions. For detail proofs of what is being presented, we refer the readers to the lecture notes of the author [L7]. 1 Gradient Estimates We begin with a discussion on an important estimate that is essential to the study of harmonic functions. In 1975, Yau [Y1] developed a maximum principle method to prove that complete manifolds with nonnegative Ricci curvature must have a Liouville property. His argument was later localized in his joint paper with Cheng [CY] and resulted in a gradient estimate for a rather general class of elliptic equations. In 1979 [L1], the maximum principle method was used by Li in proving eigenvalue estimates for compact manifolds. This method was then refined and used by many authors ([LY1], [ZY], etc) for obtaining sharp eigenvalue estimates. In 1986, Li and Yau [LY2] used a similar philosophy to prove a parabolic version of the gradient estimate for the parabolic Schrödinger equation. This philosophy has since been used by Hamilton [H1, H2, H3, H4], Chow-Hamilton [CH], Cao [Ca], Cao-Ni [CN] and many other authors to yield estimates for various non-linear parabolic equations. In a recent work [LW9] of Li and Wang, they realized that one can improve the gradient estimate of Yau and yield a sharp version of the estimate in which equality is achieved on a manifold with negative Ricci curvature. The sharpness of this was somewhat unexpected since the parabolic gradient estimate is sharp on manifolds with nonnegative Ricci curvature. In what follows, we will present the sharp gradient estimate of [LW9] for positive functions satisfying the equation f = λ f where λ 0 is a constant. Both the local and the global versions are included and various immediate consequences of the gradient estimate will also be stated. Theorem 1.1 (Yau, Li-Wang). Let M m curvature bounded from below by be a complete manifold with Ricci Ric M (m 1)K for some constant K 0. If f is a positive function defined on the geodesic ball B p (2R) M satisfying f = λ f

3 Harmonic Functions on Complete Riemannian Manifolds 191 for some constant λ 0, then there exists a constant C depending on m such that f 2 f 2 (x) (4(m 1)2 + 2ɛ)K + C((1 + ɛ 1 )R 2 + λ), 4 2ɛ for all x B p (R) and for any ɛ < 2. Moreover, if f is defined on M, then f 2 f 2 (x) (m 1)2 K (m 1)4 K λ + 2 (m 1) λ K. and λ (m 1)2 K. 4 Integrating this estimate along a geodesic joining two points in B p ( R 2 ), one derives a Harnack type inequality. Corollary 1.2. Let M m be a complete manifold with Ricci curvature bounded from below by Ric M (m 1)K for some constant K 0. If f is a positive function defined on the geodesic ball B p (2R) M satisfying f = λ f for some constant λ 0, then there exists constants C 9, C 10 > 0 depending on m such that f(x) f(y) C 9 exp(c 10 R K + λ) for all x, y B p ( R 2 ). Theorem 1.1 also allows us to recover the upper bound of Cheng [Cg1] on the bottom of the spectrum. Corollary 1.3 (Cheng). Let M m be a complete manifold with Ricci curvature satisfying Ric M (m 1)K, for some constant K 0. If we denote λ 1 (M) to be the infimum of the spectrum of the Laplacian acting on L 2 functions, then λ 1 (M) (m 1)2. 4 The Liouville theorems of Yau [Y1] and Cheng [Cg2] also follows as consequences.

4 192 Peter Li Corollary 1.4 (Yau). Let M m be a complete manifold with nonnegative Ricci curvature, then it does not admit any nonconstant, positive, harmonic functions. Corollary 1.5 (Cheng). Let M m be a complete manifold with nonnegative Ricci curvature. There exists a constant C(m) > 0, such that, for any harmonic function defined on M if we denote then s(r) = sup f(x) x B p(r) sup f (x) C R 1 s(2r). x B p(r) In particular, M does not admit any nonconstant harmonic function satisfying the growth estimate r 1 (x) f(x) 0 as x. To illustrate the sharpness of Theorem 1.1, we consider the following examples. Example 1.6. Let M m = R N m 1 be the complete manifold with the warped product metric ds 2 M = dt 2 + exp(2t) ds 2 N. A direct computation shows that the Ricci curvature on M is given by and Ric 1j = (m 1)δ 1j, for 1 j m, Ric αβ = exp( 2t) Ric αβ (m 1)δ αβ, for 2 α, β m. Here Ricαβ is the Ricci tensor on N and e 1 = t. In particular, if the Ricci curvature of N is non-negative, then Ric M (m 1). Moreover, N is Ricci flat if and only if M is Einstein with Ric M = (m 1). Let f = exp( αt) be a positive function defined on M for a constant m 1 2 α m 1. One computes that f = d2 f + (m 1)df dt2 dt = α 2 f (m 1)α f = α(m 1 α)f

5 Harmonic Functions on Complete Riemannian Manifolds 193 and f 2 = α 2 f 2. When compare with the estimate of Theorem 1.1 using K = 1 and λ = α(m 1 α), we observe that the inequality is in fact equality. Example 1.7 Let M m = H m be the hyperbolic m-space with constant curvature 1. Using the upper half-space model, H m is given by R m + = {(x 1, x 2,..., x m ) x m > 0} with metric ds 2 = x 2 m (dx dx 2 m). For x = (x 1,..., x m ) let us consider the function f(x) = xm m 1. Direct computation yields and f 2 = x 2 m m ( f i=1 x i ) 2 = (m 1) 2 x 2m 2 m = (m 1) 2 f 2. f = x 2 m = 0. m i=1 2 f x 2 i f (m 2)x m x m This implies that the gradient estimate for harmonic functions is sharp on H m.. 2 Green s Function and Parabolicity Let M m be a compact manifold of dimension m with boundary M. Let be the Laplacian defined on functions with Dirichlet boundary condition on M. Standard elliptic theory asserts that there exists a Green s function G(x, y) defined on M M \ D, where D = {(x, x) x M}, so that M G(x, y) f(y) dy = f(x) (2.1) for all functions f satisfying the Dirichlet boundary condition f M = 0. Moreover, G(x, y) = 0 for y M and x M \ M. Since both G and f satisfy Dirichlet boundary condition, after integration by parts, (2.1) becomes M y G(x, y) f(y) dy = f(x) (2.2)

6 194 Peter Li which is equivalent to saying that y G(x, y) = δ x (y), (2.3) where δ x (y) is the delta function at x. If we let f(x) = G(z, x), then (2.2) yields G(z, x) = y G(x, y) G(z, y) dy. M However, applying (2.1) to the right hand side, we obtain G(z, x) = G(x, z). (2.4) This shows that G(x, y) must be symmetric in the variables x and y. We also observe that by letting f(x) = x G(z, x) in (2.2) x G(z, x) = y G(x, y) y G(z, y) dy. M Since the right hand side is symmetric in x and z, we conclude that x G(z, x) = z G(x, z). On the other hand, z G(x, z) = z G(z, x) by the symmetry of G, we conclude that Therefore (2.2) can be written as x M x G(z, x) = z G(z, x). G(x, y) f(y) dy = M x G(x, y) f(y) dy Hence if we define the operator 1 by 1 f(x) = G(x, y) f(y) dy, M = f(x) (2.5) then (2.1) and (2.5) gives and 1 = I 1 = I, respectively.

7 Harmonic Functions on Complete Riemannian Manifolds 195 When M m is a complete, non-compact manifold without boundary, one would like to obtain a Green s function G(x, y) that possesses the properties (2.4), (2.1) and (2.5) when applied to any compactly supported function f Cc (M). For the case M = R m, it is well-known that the function { (m(m 2)ωm ) 1 r(x, y) 2 m for m 3 G(x, y) = (2ω 1 ) 1 log(r(x, y)) for m = 2, with ω m being the volume of the unit m-ball in R m, will satisfy these properties. In fact, the following example will show that these formulas, when interpreted appropriately, reflect the ideal situations for manifolds also. Example 2.1. Let us now assume that M m has a point p around which the metric is rotationally symmetric. This is equivalent to saying that if we take polar coordinates (r, θ 2,..., θ m ) around p, then ds 2 M = dr 2 + g αβ dθ α dθ β with g αβ (r) being functions of r alone. The Laplacian with respect to this coordinate system will take the form = 2 r g g r r + B p(r) where g = det(g αβ ) and Bp(r) denotes the Laplacian defined on the sphere, B p (r), of radius r centered at p. If we let A p (r) to be the area of B p (r), then g dθ = Ap (r). Since g is independent of the θ α s, and we have 1 g g r = A p(r) A p (r), = 2 r 2 + A p(r) A p (r) Let us now consider the function G(y) = r(y) 1 r + B p(r). dt A p (t) where r(y) is the distance from p to y. Direct computation gives G = 2 G r 2 = 0 + A p(r) G A p (r) r

8 196 Peter Li for y p. Moreover, if f Cc (M) is a compactly supported smooth function, then G(y) f(y) dy = G(y) f(y) dy M\B p(ɛ) = M\B p(ɛ) ɛ 1 B p(ɛ) dt A p (t) G(y) f r dθ + B p(ɛ) Note that the continuity of f implies that 1 f dθ f(p) A p (ɛ) B p(ɛ) B p(ɛ) f r dθ 1 A p (ɛ) G r f dθ B p(ɛ) as ɛ 0. Also the continuity of f and the divergence theorem imply that 1 f V p (ɛ) r dθ = 1 f dy V p (ɛ) B p(ɛ) f(p) B p(ɛ) as ɛ 0, where V p (r) is the volume of B p (r). This implies that f r dθ f(p) V p(ɛ). On the other hand, since we have ɛ 1 B p(ɛ) A p (t) m ω m t m 1, 1 dt ɛ 2 m for m 3 A p (t) m(m 2) ω m 1 log(ɛ) for m = 2. 2 ω 2 Therefore, by letting ɛ 0, the right hand side of (2.6) becomes ( ɛ dt f lim ɛ 0 A p (t) B p(ɛ) r dθ 1 ) f dθ = f(p), A p (ɛ) B p(ɛ) hence 1 M G(y) f(y) dy = f(p). In particular, this implies that one can take f dθ. (2.6) r(y) dt G(p, y) = 1 A p (t) (2.7)

9 Harmonic Functions on Complete Riemannian Manifolds 197 to be a Green s function. Note that if dt A p (t) < then by adding this constant to G, we can use the formula 1 G(p, y) = r(y) dt A p (t). (2.8) It is in the form of (2.7) and (2.8) that the Green s function in R m is given by for the cases m = 2 and m 3, respectively. The existence of a Green s function on a general complete manifold was first proved by Malgrange [M]. However, for the purpose of application, a constructive argument is a key step in getting the appropriate estimates. The first constructive proof was published in [LT2], and we will give a brief description below. Let {Ω i } i=1 be a compact exhaustion of M, such that Ω i Ω j for i < j, i=1ω i = M, and each Ω i is a sufficiently smooth compact subdomain of M. For each i, let G i (x, y) be the symmetric, positive, Green s function with Dirichlet boundary condition of Ω i. Moreover, G i (p, y) { (m(m 2)ωm ) 1 r 2 m (p, y) for m 3 (2ω 2 ) 1 log r(p, y) for m = 2, as y p. The construction argument of [LT2] asserts that if M m admits a positive Green s function, G(p, y), then G can be obtained by G(p, y) = lim i G i (p, y). Theorem 2.1. Let M m be a complete manifold without boundary. There exists a Green s function G(x, y) which is smooth on (M M) \ D satisfying properties (2.1), (2.4), and (2.5). Moreover, G(x, y) can be taken to be positive if and only if there exists a positive superharmonic function f on M \ B p (R) with the property that lim inf f(x) < inf f(x). x x B p(r) The existence and nonexistence of a positive Green s function divides the the class of complete manifolds into two categories. In general the methods in dealing with function theory on these manifolds are different. Hence it is important to understand the difference in the two categories.

10 198 Peter Li Definition 2.2. A complete manifold is said to be parabolic if it does not admit a positive Green s function. Otherwise it is said to be nonparabolic. As pointed out in Theorem 2.1, a manifold is nonparabolic if and only if there exists a positive superharmonic function whose infimum is achieved at infinity. This property can be localized at any unbounded component at infinity. Definition 2.3. An end, E, with respect to a compact subset Ω M is an unbounded connected component of M \ Ω. The number of ends with respect of Ω, denoted by N Ω (M), is the number of unbounded connected component of M \ Ω. It is obvious that if Ω 1 Ω 2, then N Ω1 (M) N Ω2 (M). Hence if {Ω i } is a compact exhaustion of M, then N Ωi (M) is a monotonically nondecreasing sequence. If this sequence is bounded, then we say that M has finitely many ends. In this case, we denote the number of ends of M by N(M) = max i N Ω i (M). One readily checks that this is independent of the compact exhaustion {Ω i }. In fact, it is also easy to see that there must be an i 0 such that N(M) = N Ωi (M), for all i i 0. Hence for all practical purposes, we may assume that M \ B p (R 0 ) has N(M) number of unbounded connect components, for some R 0. In general, when we say that E is an end we mean that it is an end with respect to some compact subset Ω. In particular, its boundary E is given by Ω Ē. Definition 2.4. An end E is said to be parabolic if it does not admit a positive harmonic function f satisfying f = 1 on E and lim inf f(y) < 1, y E( ) where E( ) denotes the infinity of E. Otherwise, E is said to be nonparabolic and the function f is said to be a barrier function of E. With this notion, we can also count the number of nonparabolic (parabolic) ends as in the definition of N(M). Definition 2.5. We denote N 0 (M) to be the number of nonparabolic ends of M, and N (M) to be the number of parabolic ends of M. Observe that by addition and multiplication of constants, we may assume that the function f in Definition 2.4 satisfied f E = 1 and lim inf f(y) = 0. (2.9) y E( )

11 Harmonic Functions on Complete Riemannian Manifolds 199 Also note that if E is a nonparabolic end of M, then by extending f to be identically 1 on (M \ Ω) \ E, it can be used to construct a positive Green s function on M. Hence, M is nonparabolic if and only if M has a nonparabolic end. Of course, it is possible for a nonparabolic manifold to have many parabolic ends. Let us also point out that E being nonparabolic is equivalent in saying that E has a positive Green s function with Neumann boundary condition. If E is parabolic, one can also construct a barrier function g which is harmonic on E and have the properties that g E = 0 and sup g =. (2.10) y E In the case when M is nonparabolic, it follows that there is a unique minimal positive Green s function. The following estimate for the minimal Green s function of Li-Tam [LT7] gives a necessary condition for the manifold to be nonparabolic. Proposition 2.6 (Li-Tam). Let M be a complete manifold. If M is nonparabolic then for any point p M dt <, (2.11) A p (t) where A p (r) denotes the area of B p (r). Green s function, then for all r > 1. r 1 dt A p (t) 1 sup y B p(1) G(p, y) Moreover, if G(p, y) is the minimal inf y B p(r) G(p, y) It is natural to ask if the condition (2.11) is also sufficient for nonparabolicity. In many cases, with extra geometric assumptions on the manifold, one can show that (2.11) indeed implies nonparabolicity. We will refer the readers to [LT7] for additional information. The following proposition gives a criterion for an end to be nonparabolic. It was first proved by Cao, Shen and Zhu in [CSZ] for minimal submanifolds. However, their arugment can be generalized to the following context. Proposition 2.7 (Cao-Zhen-Zhu). Let E be an end of a complete Riemannian manifold. Suppose for some ν 1 and C > 0, E satisfies a Sobolev type inequality of the form ( ) 1 u 2ν ν C u 2 E for all compactly supported function u Hc 1,2 (E) defined on E, then E must either have finite volume or be nonparabolic. On the other hand, we also have the following proposition. E

12 200 Peter Li Proposition 2.8. Let M be a complete Riemannian manifold. Suppose for some ν > 1 and C > 0, M satisfies a Sobolev type inequality of the form ( E u 2ν ) 1 ν C E u 2 for all compactly supported function u H 1,2 c (M). There exists a constant C 1 > 0 depending only on C and ν, such that, the volume of a geodesic ball of radius R centered at p M must satisfy V p (R) C 1 R 2ν ν 1. In particular, each end of M must have at least R 2ν ν 1 volume growth. Combining Propositions 2.7 and 2.8, we obtain the following corollary for the Sobolev type inequality with ν > 1. The case when ν = 1 is just the Dirichlet Poincaré inequality and in that case, it is possible to have a finite volume end given by a cusp. Corollary 2.9. Let E be an end of a complete Riemannian manifold. Suppose for some ν > 1 and C > 0, E satisfies a Sobolev type inequality of the form ( E u 2ν ) 1 ν C E u 2 for all compactly supported function u Hc 1,2 (E) defined on E, then E must be nonparabolic. 3 Heat Kernel Estimates and Mean Value Inequality In this section, we will present the estimates for positive solutions of the heat equation proved by Li and Yau in [LY2]. In particular, upper bound for the fundamental solution of the heat equation will be established for manifolds with Ricci curvature bounded from below. The gradient estimate given in Theorem 3.1 has fundamental importance in studying parabolic equations. There are many subsequent development of similar type estimates for other nonlinear partial differential equations. However, we will not discuss these directions since they are out of the scope of these notes. The purpose of this section is to present the essential estimates that are necessary for our purpose and for the proof of the the mean value inequality. Theorem 3.1 (Li-Yau). Let M m be a complete manifold with boundary. Assume that p M and R > 0 so that the geodesic ball B p (4R) does not intersect the boundary of M. Suppose the Ricci curvature of M on B p (4R) is bounded from below by Ric M (m 1)K

13 Harmonic Functions on Complete Riemannian Manifolds 201 for some constant K 0. If f(x, t) is a positive solution of the equation ( ) f(x, t) = 0 t on M [0, T ], then for any 1 < α 2, the function h(x, t) = log f(x, t) satisfies the estimate h 2 αh t m 2 α2 t 1 + C 1 (α 1) 1 (R 2 + K) on B p (2R) {t} for 0 t T, where C 1 is a constant depending only on m. This estimate takes a much simpler form when the function is defined globally on a manifold with nonnegative Ricci curvature. In fact, the inequality is sharp on R m. Corollary 3.2. Let M m be a complete manifold with nonnegative Ricci curvature. If f(x, t) is a positive solution of the heat equation ( ) f(x, t) = 0 t on M [0, ), then on M [0, ). f 2 f 2 f t f m 2t Theorem 3.1 implies a Harnack type inequality, which is extremely useful in the study of parabolic equations. One simply integrate the gradient estimate along a suitably chosen curve in M R joining the points (x, t 1 ) and (y, t 2 ). Corollary 3.3. Under the same hypotheses of Theorem 3.1, f(x, t 1 ) f(y, t 2 ) ( t2 t 1 ) mα 2 ( α r 2 ) (x, y) exp 4(t 2 t 1 ) + C 1(α 1) 1 (R 2 + K) ((t 2 t 1 ) for any x, y B p (R) and 0 < t 1 < t 2 T, where r(x, y) is the geodesic distance between x and y. Using Theorem 3.1, we can derive an upper bound on the fundamental solution of the heat equation. Theorem 3.4 (Li-Yau). Let M m be a complete manifold and H(x, y, t) denotes the minimal symmetric heat kernel defined on M M (0, ) with the properties that ( y ) H(x, y, t) = 0 t

14 202 Peter Li and lim H(x, y, t) = δ x(y). t 0 For any p M, ɛ > 0, R > 0 and t R2 4, if the Ricci curvature of M on B p(4r) is bounded from below by Ric M (m 1)K then there exists constants C 1 > 0 depending only on m and ɛ and C 2 > 0 depending only on m, such that, H(x, y, t) C 1 exp( λ 1 t) V 1 2 x ( t) V 1 2 y ( t) ( exp r2 (x, y) 4(1 + 2ɛ)t + ) C 2 (R 2 + K)t for any x, y B p (R) and t R2 4. Another important inequality is the mean value inequality for positive subharmonic functions. The classical mean value property for harmonic functions in R m asserts that the average value of a harmonic function on a ball is given by the value of the function at the center point, i.e., Vp 1 (R) f = f(p). B p(r) If f is a nonnegative subharmonic function on R m, it must satisfy the inequality Vp 1 (R) f f(p). B p(r) The validity of such inequality on a manifold is important in the study of nonlinear analysis and certainly for the study of harmonic functions. Of course, the precise form of the inequality might not be the same as the one in R m, but one expects a mean value inequality of the form C Vp 1 (R) f f(p), B p(r) where C > 0 is a constant that may depend on M. By using the Sobolev inequality and Moser s iteration method, one can obtain the mean value inequality in the above form. However, the constant C will depend on the Sobolev constant. One can then estimate the Sobolev constant by estimating the isoperimetric inequality as in [Cr] (also see [L4]), but the estimate will depend on the volume V p (R) and other geometric quantities. On the other hand, it is often desirable that the constant C is independent of the volume so that the left hand side is indeed the average value of f. In [LS], Li and Schoen proved a mean value inequality where the constant C only depends on the lower bound of the Ricci curvature, m

15 Harmonic Functions on Complete Riemannian Manifolds 203 and R. Their argument works for nonnegative functions satisfying the differential inequality f λ f (3.1) for 0 λ λ 1 (B p (R)) where λ 1 (B p (R)) is the first Dirichlet eigenvalue of B p (R). In other applications, it is useful to derive a mean value inequality for nonnegative function f satisfying (3.1) without any restriction on λ. This was done by Li and Tam [LT5] using the upper bound of the heat kernel. They actually proved a mean value inequality for nonnegative subsolutions of the heat equation, which includes subsolutions of (3.1) as a special case. We would like to point out that the value of the mean value constant is also of significance. One may refer to [LW2] for further details. Theorem 3.5 (Li-Tam). Let M m be a complete noncompact Riemannian manifold with boundary. Let p M and R > 0 be such that the geodesic ball B p (2R) does not intersect the boundary of M. Suppose g(x, t) is a nonnegative function defined on B p (2R) [0, T ] for some 0 < T R2 4 satisfying the differential inequality g g t 0. If the Ricci curvature of B p (2R) is bounded by Ric M (m 1)K for some constant K 0, then for any q > 0, there exists positive constants C 1 and C 2 depending only on m and q such that for any 0 < τ < T, 0 < δ < 1 2, and 0 < η < 1 2, sup g q V (K, 2R) C 1 B p((1 δ)r) [τ,t ] V p (R) ( R ) K + 1 exp(c 2 K T ) ( 1 δr + 1 ) m+2 T ds ητ (1 η)τ B p(r) g q (y, s) dy, where V (K, r) is the volume of the geodesic ball of radius r in m-dimensional constant curvature space form with constant sectional curvature K. Corollary 3.6 (Li-Schoen, Li-Tam). Let M m be a complete manifold with boundary and p be a fixed point in M. Suppose R > 0 be such that B p (2R) does not intersect the boundary of M and assume that the Ricci curvature of M on B p (2R) satisfies the bound Ric M (m 1)K for some constant K 0. Let 0 < δ < 1 2, q > 0, and λ 0 be fixed constants. Then there exists a constant C > 0 depending only on δ, q, λ R 2, m, and R K, such

16 204 Peter Li that, for any nonnegative function f defined on B p (2R) satisfying the differential inequality f λ f, we have sup B p((1 δ)r) f q C Vp 1 (R) f q (y) dy. B p(r) 4 Harmonic Functions and Ends A globally defined harmonic functions can be constructed by extending the barrier functions given by (2.9) and (2.10) defined on each of the ends of M. The construction was first proved by Tam and Li in [LT1] for manifolds with nonnegative sectional curvature near infinity. They later gave a construction for arbitrary complete manifolds in [LT6]. In [STW], Sung, Tam, and Wang presented the construction in a more systematic manner and gave general criteria on the extendability of a harmonic function that is defined only near infinity. In particular, these harmonic functions will reflect the geometry and topology of a complete manifold. Theorem 4.1 (Li-Tam). Let M be a complete manifold that is nonparabolic. There exists spaces of harmonic functions K 0 (M) and K (M), with (possibly infinite) dimensions given by k 0 (M) and k (M), respectively, such that k 0 (M) = N 0 (M) and In particular, k (M) = N (M). k 0 (M) + k (M) = N(M). Moreover, K 0 (M) is a subspace of the space of bounded harmonic functions with finite Dirichlet integral on M, and K (M) is spanned by a set of positive harmonic functions. A similar construction also gives a corresponding theorem for parabolic manifolds. Theorem 4.2. Let M be a complete manifold that is parabolic. There exists a space of harmonic functions K(M), with (possibly infinite) dimension given by k(m), such that k(m) = N(M), Moreover, K(M) is spanned by a set of harmonic functions which are bounded either from above or below when restricted on each end of M. We will now give an application of this theory to study the infinity structure of a complete manifold whose Ricci curvature is almost nonnegative. In particular,

17 Harmonic Functions on Complete Riemannian Manifolds 205 we will give an estimate on the number of ends of such a manifold. When M has nonnegative Ricci curvature everywhere, this argument also recovers the splitting theorem of Cheeger and Gromoll [CG]. It was proved by Li and Tam [LT6] that if we assume that the Ricci curvature of M is bounded from below by Ric M (x) (m 1)k(r(x)) for some nonincreasing function k(r) 0 satisfying the property 1 k(r) r m 1 dr <, then N(M) is finite and can be estimated. We will first state a lemma that is very useful in estimating dimensions of linear spaces of sections. The lemma was first proved by Li in [L2] and it will be used in many of the theorems stated in these notes. Lemma 4.3 (Li). Let Ω M m be a compact subset of a complete manifold M. Suppose T Ω is a rank n vector bundle with fibers given by a vector space E endowed with an inner product,. Let F be a finite dimensional linear space of sections of T. Then there exists a section f F such that dim F Ω f 2 n V (Ω) sup f 2, Ω where dim F is the dimension of F and f is the norm with respect to the inner product of E. Theorem 4.4 (Li-Tam). Let M m be a complete manifold and p M be a fixed point such that Ric M (x) k(r(x)) where r(x) denotes the distance to the point p. Suppose that k : [0, ) [0, ) is a nonincreasing continuous function such that r m 1 k(r)dr <. Then there exists a constant C(m, k) depending only 0 on m and k, such that, the number of ends of M is bounded by N(M) C(m, k). Corollary 4.5. Let M m be a complete manifold. Suppose Ω B p (R 0 ) is a compact subset of M such that the Ricci curvature of M is nonnegative on M \ Ω. Then there exists a constant C > 0 depending only on m, R 0 and the lower bound of the Ricci curvature on B p (R 0 ), such that, M has at most C number of ends. Moreover, if M has nonnegative Ricci curvature everywhere then M either has only one end, or it must be isometrically a cylinder M = R N given by the product metric, where N is a compact manifold with nonnegative Ricci curvature.

18 206 Peter Li 5 Stability of Minimal Hypersurfaces In this section, we will give applications of harmonic functions to the study of complete minimal hypersurfaces. Let N m+1 be a complete manifold with nonnegative Ricci curvature. Suppose M m is a complete minimal hypersurface in N. If A 2 denotes the square of the length of the second fundamental form of M and Ric N (ν, ν) is the Ricci curvature of N in the direction of the unit normal ν to M, then M being stable in N is characterized by the stability inequality [SY] M ψ 2 A 2 + ψ 2 Ric N (ν, ν) ψ 2 (5.1) M M for any compactly supported function ψ Hc 1,2 (M). Geometrically the stability inequality is derived from the second variation formula for the volume functional under normal variations. Hence a stable minimal hypersurface is not only a critical point of the volume functional but its second derivative is nonnegative with respect to any normal variations. The elliptic operator associated to the stability inequality is given by L = + A 2 + Ric N (ν, ν). The stability of M is equivalent to the fact that the operator L is nonnegative. We say that M has finite index when the operator L has only finitely many negative eigenvalues. This has the geometric interpretation that there is only a finite dimensional space of normal variations violating the stability inequality. The study of stable minimal hypersurfaces can be viewed as an effort of proving a generalized Bernstein s theorem. Bernstein first established that an entire minimal graph - a minimal hypersurface which is given by a graph of a function defined on R 2. - in R 3 must be a plane. The validity of Bernstein theorem in higher dimension was established for entire minimal graphs in R m+1 for m 7 by Simons [S], and many other authors, such as Fleming [Fl], Almgren [A], and DeGiorgi [De], for the lower dimensional cases. Counter-examples for m 8 was found by Bombieri, DeGiorgi, and Guisti [BDG]. Since entire minimal graphs are area minimizing, a natural question is to ask if a Bernstein type theorem is valid for stable minimal hypersurfaces in R m+1. In 1979, do Carmo and Peng [dcp] proved that a complete, stable, minimally immersed hypersurface M in R 3 must be planar. At the same time, Fischer- Colbrie and Schoen [FS] independently showed that a complete, stable, minimally immersed hypersurface M in a complete 3-dimensional manifold N with nonnegative scalar curvature must be either conformally a plane R 2 or conformally a cylinder R S 1. For the special case when N is R 3, they also proved that M must be planar. In 1984, Gulliver [G1] studied a yet larger class of submanifolds in R 3. He proved that a complete, oriented, minimally immersed hypersurface with finite index in R 3 must have finite total curvature. In particular, after applying Huber s theorem, one concludes that the hypersurface must be conformally equivalent to

19 Harmonic Functions on Complete Riemannian Manifolds 207 a compact Riemann surface with finitely many punctures. The same result was also independently proved by Fischer-Colbrie in [F]. In addition, she also proved that a complete, oriented, minimally immersed hypersurface with finite index in a complete 3-dimensional manifold with nonnegative scalar curvature must be conformally equivalent to a compact Riemann surface with finite punctures. Shortly after, Gulliver [G2] improved the result of Fischer-Colbrie and showed that if the ambient manifold has nonnegative scalar curvature then a minimal hypersurface with finite index must have quadratic area growth, finite topological type, and the length of the second fundamental form must be square integrable. Indeed, a complete surface with quadratic area growth and finite topological type must be conformally equivalent to a compact Riemann surface with finitely many punctures. In 1997, Cao, Shen, and Zhu [CSZ] considered the high dimensional cases of the theorem of do Carmo-Peng and Fischer-Colbrie-Schoen. They proved that a complete, oriented, stable, minimally immersed hypersurface M n in R n+1 must have only one end. This theorem was generalized by Li and Wang [LW6] (see Theorem 5.6), where they showed that a complete, oriented, minimally immersed hypersurface M n in R n+1 with finite index must have finitely many ends. In a later paper [LW8], Li and Wang also generalized their theorem to minimal hypersurfaces with finite index in a complete manifold with nonnegative sectional curvature. In this section, we will first present the relationship between harmonic functions and stability of minimal hypersurfaces. Taken the point of view of function theory, one readily recovers most of the two dimensional results stated earlier. We also present higher dimensional results that can be obtained using this point of view. Note that when the ambient manifold is Euclidean space and n 3, then M is non-parabolic by applying Proposition 2.8. The key issue is the validity of Sobolev inequality proved by Michael and Simon [MS] in the form ( M ) m 2 u 2m m m 2 C M u 2 on any minimal submanifolds of R n. However, when N is only assumed to have nonnegative Ricci (or sectional) curvature, then M can be parabolic as in the case of the cylinder M = R P in N = R 2 P. The next theorem states that the case M being parabolic is a very special situation. Proposition 5.1. Let M m be a complete, minimally immersed, stable, hypersurface in a manifold, N m+1, with nonnegative Ricci curvature. If M is parabolic, then it must be totally geodesic in N. Moreover, the Ricci curvature Ric N (ν, ν) of N in the normal direction to M also vanishes, and M must have nonnegative scalar curvature. Proposition 5.1 reduces our study of stable minimal hypersurfaces to the nonparabolic case. The following lemma of Schoen and Yau [SY] illustrates the role of harmonic functions in the stability inequality.

20 208 Peter Li Lemma 5.2 (Schoen-Yau). Let M m be a complete, minimally immersed, stable, hypersurface in N m+1. Suppose N has nonnegative sectional curvature and u is a harmonic function defined on M. Then the inequality 1 φ 2 A 2 u φ 2 u 2 φ 2 u 2 m M m 1 M M holds for any compactly supported, nonnegative function φ Hc 1,2 (M). In the case when m = 2, Schoen and Yau [SY] observed that the stability inequality can be rewritten using the scalar curvature of N, denoted by S N. One also observed that the stability inequality can be lifted to the universal covering of M, and as a corollary of Proposition 5.1 and Lemma 5.2, one readily recovers the theorems of Fischer-Colbrie and Schoen [FS] (also by do Carmo and Peng [dcp] for the special case when N = R 3 ). Theorem 5.3 (Fischer-Colbrie-Schoen, do Carmo-Peng). Let M 2 be an oriented, complete, stable, minimal hypersurface in a complete manifold N 3 with nonnegative scalar curvature. Then M must be conformally equivalent to either the complex plane C or the cylinder R S 1. If M is conformally equivalent to the cylinder and has finite total curvature, then it must be totally geodesic and the scalar curvature of N along M must be identically 0. Corollary 5.4 (Fischer-Colbrie-Schoen). Let M 2 be an oriented, complete, stable, minimal hypersurface in a complete manifold N 3 with nonnegative Ricci curvature. Then M must be totally geodesic in N and the Ricci curvature of N in the normal direction to M must be identically zero along M. Moreover, M is either (1) conformally equivalent to the complex plane C; or (2) isometrically the cylinder R S 1. Moreover, if N = R 3, then M must be planar in R 3. Let us now consider the case when m > 2. We assume that N m+1 is a complete manifold with nonnegative sectional curvature. Theorem 5.5 (Li-Wang). Let M m be a complete, stable, minimally immersed hypersurface in N m+1. Suppose N is a complete manifold with nonnegative sectional curvature. If M is parabolic, then it must be totally geodesic and has nonnegative sectional curvature. In particular, M either has only one end, or M = R P with the product metric, where P is compact with nonnegative sectional curvature. If M is non-parabolic, then it must only have one non-parabolic end. In this case, any parabolic end of M must be contained in a bounded subset of N. Corollary 5.6. Let M m be a complete, properly immersed, stable, minimal hypersurface in N m+1. Suppose N is a complete manifold with nonnegative sectional curvature. Then either (1) M has only one end; or

21 Harmonic Functions on Complete Riemannian Manifolds 209 (2) M = R P with the product metric, where P is compact with nonnegative sectional curvature, and M is totally geodesic in N. This corollary includes the theorem of Cao, Shen, and Zhu [CSZ], where they gave the first structural description of a stable minimal hypersurface in R m+1 for m 3. Using harmonic functions, the number of ends can be seen to be bounded for the minimal hypersurface has finite index. Theorem 5.7 (Li-Wang). Let M m be a minimally immersed hypersurface in R m+1 with m 3. If M has finite index then there exist a constant C > 0 depending on M such that N(M) C. 6 Polynomial Growth Harmonic Functions In this section, we will consider polynomial growth harmonic functions on a complete manifold. Recall that any polynomial growth harmonic function in R m is necessarily a polynomial with respect to the variables in rectangular coordintates. Hence the space of polynomial growth harmonic functions of at most order d are given by the space of harmonic polynomials of degree at most d. In particular, these spaces are all finite dimensional. Definition 6.1. Let r(x) is the distance from x to a fixed point p M. We denote H d (M) = {f f = 0, and f(x) C r d (x) for some constant C > 0}, to be the vector space of all polynomial growth harmonic functions defined on M of order at most d. We also denote the dimension of this vector space H d (M) by h d (M). Using this notation, one computes that ( h d (R m m + d 1 ) = d ) + ( ) m + d 2. d 1 On the other hand, Cheng s theorem (Corollary 1.5) asserts that if M has nonnegative Ricci curvature, then h d (M) = 1 for all d < 1. In view of this, Yau conjectured ([Y3] problem 48) that h d (M) must be finite dimensional if M has nonnegative Ricci curvature. He also raised a similar questions for holomorphic sections of vector bundles. In 1989, Li and Tam [LT3] proved that h 1 (M) m + 1 if M has nonnegative Ricci curvature. Note that this estimate is sharp and it is achieved by R m.

22 210 Peter Li Theorem 6.2 (Li-Tam). Let M be a complete manifold with nonnegative Ricci curvature. Suppose the volume growth of M satisfies lim sup R n V p (R) < R for some n m, then h 1 (M) n + 1 m + 1. In fact, Li [L5] showed that if M is Kähler and has nonnegative holomorphic bisectional curvature then equality holds if and on if M = C n with 2n = m. The real case of this theorem was later proved by Cheeger, Colding, and Minicozzi [CCM]. They showed that if M has nonnegative Ricci curvature and h 1 (M) = m + 1, then M = R m. In the case of Yau s conjecture for polynomial growth harmonic functions, it was first proved by Colding and Minicozzi [CM1-3] and they gave the estimate h d (M) C d m 1 for some constant C > 0 depending on M. In [L6], Li gave a simplified proof of this estimate that holds for a much larger class of manifolds, which we will now give a brief description. Definition 6.3. A manifold is said to satisfy a volume comparison condition (V µ ) for some µ > 1, if for any point x M and any real numbers 0 < R 1 R 2 < the volume of the geodesic balls centered at x must satisfy V x (R 2 ) V x (R 1 ) ( R2 Definition 6.4. A manifold is said to satisfy a mean value inequality (M) if there exists a constant C 0 > 0 such that for any x M and R > 0, the inequality f 2 (x) C 0 Vx 1 (R) f 2 (y) dy R 1 B x(r) is valid for all nonnegative subharmonic function f. Note that if M has nonnegative Ricci curvature then M satisfies condition (V m ) by the Bishop volume comparison theorem, and condition (M), by the theorem of Li-Schoen [LS] (Corollary 3.6). Theorem 6.5 (Li). Let M m be a complete manifold satisfying conditions (V µ ) and (M). Suppose E is a rank-n vector bundle over M. Let S d (M, E) Γ(E) be a linear subspace of sections of E, such that, all u S d (M, E) satisfy (a) u 0, and (b) u (x) O(r d (x)) as x. ) µ.

23 Harmonic Functions on Complete Riemannian Manifolds 211 Then the dimension of S d (M, E) is finite. Moreover, for all d 1, there exists a constant C > 0 depending only on µ such that dim S d (M, E) n C C 0 d µ 1. In particular, we recover the theorem of Colding and Minicozzi and also answered Yau s question on sections of certain class of vector bundles. Corollary 6.6 (Colding-Minicozzi). Let M m be a complete Riemannian manifold with nonnegative Ricci curvature. There exists a constant C > 0 depending only on m, such that, dimension of the H d (M) is bounded by for all d 1. h d (M) C d m 1 In fact, finite dimensionality of H d (M) can be obtained by substantially relaxing both the volume comparison condition and the mean value inequality condition. However, the order of dependency in d will not be sharp. Definition 6.7. A complete manifold M is said to satisfy a weak mean value inequality (WM) if there exists constants C 0 > 0 and b > 1 such that, for any nonnegative subharmonic function f defined on M, it must satisfy f(x) C 0 Vx 1 (R) f(y) dy. for all x M and R > 0. B x(br) Theorem 6.8 (Li). Let M be a complete manifold satisfying the weak mean value property (WM). Suppose that the volume growth of M satisfies V p (R) = O(R µ ) as R for some point p M. Then H d (M) is finite dimensional for all d 0 and dim H d (M) C 0 (2b + 1) (2d+µ). Definition 6.9. A complete manifold M is said to satisfy a Sobolev inequality (S) if there exist constants C 1 > 0 and µ > 2, such that, for all p M, R > 0, and for all f H c 1,2(M), we have ( ) µ 2 f 2µ µ µ 2 B p(r) C 1 V p (R) 2 µ B p(r) (R 2 f 2 + f 2 ). Corollary Let M be a complete manifold with property (S). Then dim H d (M) < C β d for all d 0 for some constant C 1 > 0 and β > 1 depending only on C 0, µ, and b. If we assume that M m has nonnegative sectional curvature then Li and Wang [LW3] proved that the quantity d i=1 hi (M) has an upper bound that is asymptotically sharp as d.

24 212 Peter Li Theorem 6.11 (Li-Wang). Let M m be a complete manifold with nonnegative sectional curvature. Let 0 α 0 ω m be a constant such that lim inf r r m V p (r) = α 0. If h d = dim H d (M) denotes the dimension of the space of polynomial growth harmonic functions of at most degree d, then lim inf d d (m 1) h d 2α 0. (m 1)! ω m Moreover, the equality holds if and only if M = R m. lim inf d d (m 1) h d 2 = (m 1)! Theorem 6.1 and Theorem 6.11 indicated that on a complete manifold with nonnegative sectional curvature h d (M) is bounded from above by h d (R m ) when d 1 or when d. One conjectures that perhaps the inequality h d (M) h d (R m ) is valid for all d. It is also interesting to point out that this is not true when the assumption is relaxed to nonnegative Ricci curvature, as pointed out by an example of Donnelly [D]. Similar studies on solutions of general elliptic operators on R m can also be found in the works of Avellaneda-Lin [AL], Lin [Ln], Moser-Struwe [MrS], and Li-Wang [LW4-5]. 7 Massive Sets and the Structure of Harmonic Maps In this section, we will introduce the notion of d-massive sets. The notion of 0-massive set was first introduced by Grigor yan [G] where he established the relationship between massive sets and harmonic functions. While it is not clear if such a relationship exists for d > 0, but the notion of d-massive sets have important geometric and analytic implications. Definition 7.1. For any real number d 0, a d-massive set Ω is a subset of a manifold M that admits a nonnegative, subharmonic function f defined on Ω with the boundary condition f = 0 on Ω, and satisfying the growth property f(x) C r d (x) for all x Ω and for some constant C > 0. The function f is called the potential function of Ω. We also denote m d (M) to be the maximum number of disjoint d-massive sets admissible on M. The following theorem was proved by Grigor yan [G].

25 Harmonic Functions on Complete Riemannian Manifolds 213 Theorem 7.2 (Grigor yan). Let M m be a complete Riemannian manifold. The maximum number of disjoint 0-massive sets admissible on M is given by the dimension of the space of bounded harmonic functions on M, i.e., m 0 (M) = h 0 (M). As pointed out earlier, there are no direct relationship between h d (M) and m d (M) for d > 0. However, Li and Wang [LW1, LW3] discovered that there are parallel theories on the two numbers that provide a philosophical connection. This is demonstrated by the following theorem that mirrors Theorem 6.6 when applied to polynomial growth harmonic functions. Theorem 7.3 (Li-Wang). Let M m be a complete manifold satisfying conditions (V µ ) and (M). For all d 1, there exists a constant C > 0 depending only on µ such that m d (M) C C 0 d µ Similarly, we also have the following finiteness theorem that mirrors Theorem Theorem 7.4 (Li-Wang). Let M be a complete manifold satisfying the weak mean value property (WM). Suppose that the volume growth of M satisfies V p (R) = O(R µ ) as R for some point p M. Then m d (M) C 0 (2b + 1) (2d+µ). One can also prove a sharp estimate when M = R 2. Theorem 7.5 (Li-Wang). On R 2, for all d > 0. m d (R 2 ) 2d We will now apply the notion of massive sets to study the structure of the image of a harmonic map into a Cartan-Hadamard manifold. More specifically, Li and Wang [LW1] developed this theory that applies to the target being a strongly negatively curved Cartan-Hadamard manifold, or when it is a 2-dimensional visibility manifold. The connection between massive sets and harmonic maps into Cartan-Hadamard manifold follows from the fact that the pullback of a convex function by harmonic map is a subharmonic function. Since any Busemann function on a Cartan-Hadamard manifold is convex, the supports of their pullback by a harmonic map yield many massive sets. Throughout the remaining of this section we shall assume that N is a Cartan- Hadamard manifold, namely, N is simply connected and has nonpositive sectional curvature. It is well known that N can be compactified by adding a sphere at

26 214 Peter Li infinity S (N). The resulting compact space N = N S (N) is homeomorphic to a closed Euclidean ball. Two geodesic rays γ 1 and γ 2 in N are called equivalent if their Hausdorff distance is finite. Then the geometric boundary S (N) is simply given by the equivalence classes of geodesic rays in N. A sequence of points {x i } in N converges to x S (N) if for some fixed point p N, the sequence of geodesic rays {px i } converges to a geodesic ray γ x. In this case, we say γ is the geodesic segment px joining p to x. Recall that a subset C in N is strictly convex if any geodesic segment between any two points in C is also contained in C. For a subset K in N, the convex hull of K, denoted by C(K), is defined to be the smallest strictly convex subset C in N containing K. The convex hull can also be obtained by taking the intersection of all convex sets C N containing K. When N is a Cartan-Hadamard manifold, there is only one geodesic segment joining a pair of points in N. In this case, there is only one notion of convexity, and we will simply say a set is convex when it is a strictly convex set. For the purpose of this article, we will need a notion of convexity for N. Since a geodesic line is a geodesic segment joining the two end points in S (N), it still makes sense to talk about geodesics joining two points in N. However, it is generally not true that any two points in S (N) can always be joined by a geodesic segment given by a geodesic line, as indicated by two non-antipodal points in S (R n ). If every pair of points in S (N) can be joined by a geodesic line in N, then N is said to be a visibility manifold. This class of manifolds was extensively studied by Eberlein and O Neill [EO]. A typical example of a visibility manifold is a Cartan-Hadamard manifold with sectional curvature bounded from above by a negative constant a < 0. To remedy the situation when N is not a visibility manifold, we defined a generalized notion of geodesic segment joining two points at infinity. Definition 7.6. A geodesic segment γ joining a pair of points x and y in S (N) is the limiting set of a sequence of geodesic segments {γ i } in N with end points {x i } and {y i } such that x i x and y i y. We will denote γ by xy. Observe that if xy S (N) = {x, y}, then xy must be a geodesic line in N and hence a geodesic segment in the traditional sense. For the case of two non-antipodal points in S (R 2 ), the shortest arc on S 1 = S (R 2 ) joining the two points will be the geodesic segment in the sense defined above. If the two points are antipodal in S (R 2 ), say the northpole and the southpole, then there are infinitely many geodesic segments joining them. Each vertical line is a geodesic segment in the genuine sense. Also, both arcs on S 1 joining the two poles are geodesic segments joining them. Using this definition, for a pair of points in S (N), it is possible to have more than one geodesic segments joining them. The convexity we will define will be in the sense of strictly convex. Definition 7.7. A subset C of N is a convex set if for every pair of points in C, any geodesic segment joining them is also in C.

27 Harmonic Functions on Complete Riemannian Manifolds 215 Definition 7.8. For a subset A in N, we define its convex hull C(A) to be the smallest convex subset of N containing A. In what follows, when we say that a subset is closed, we mean that it is closed in N unless otherwise noted. In general, we denote the closure for a subset A in N by Ā. For a given sequence of closed subsets {A i} decreasing to A, it is natural to ask whether the convex hull of A i in N decreases to the convex hull of A. For this purpose, we introduce the following definition. Definition 7.9. A Cartan-Hadamard manifold N is said to satisfy the separation property if for every closed convex subset A in N and every point p not in A, there exists a closed convex set C properly containing A and separating p from A, i.e., A C, A S (N) is contained in the interior of C S (N) and p is not in C. For a two-dimensional visibility manifold or a Cartan-Hadamard manifold with constant negative curvature, it is easy to check that the separation property holds. In fact, for a point p not in the closed convex set A, pick up a point q A such that r(p, q) = r(p, A). Then the convexity of A and the first variation formula imply that for z A, (zq, qp) π/2. Let x be the midpoint of the geodesic segment between p and q, and C = {y N : (yx, xp) π/2}. Then C is closed, convex as C is evidently totally geodesic and C properly separates p from A. Proposition A Cartan-Hadamard manifold N satisfies the separation properly if and only if for every closed subset A and monotone decreasing sequence of closed subsets {A i } in N such that i=1 A i = A, then i=1 C(A i ) = C(A). According to our definition of convex hull, it is possible that C(K) S (N) is a much bigger set than K S (N). In fact, if we consider K to be the y-axis in R 2, then K S (R 2 ) consists of the two poles in S 1. However, C(K) = R 2 because every line given by x = constant is a geodesic joining the two poles of S 1. Hence, C(K) S (R 2 ) = S 1. On the other hand, if we assume in addition that N satisfies the following separation property at infinity, then C(K) S (N) = K S (N). Definition Let N be a Cartan-Hadamard manifold. N is said to satisfy the separation property at infinity if for any closed subset A of S (N) and any point p S (N) \ A, there exists a closed convex subset C in N such that A is contained in the interior of C S (N) and p not in C.