HarmonicFunctionson CompleteRiemannianManifolds...functions on complete Riemannian manifolds. Since the seminal work of Yau [Y1] in 1985, where he proved a Liouville theorem for positive

* · Vol. I · ?–?

Harmonic Functions onComplete 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 developedtechniques in the subject and also geometric applications that followed.

2000 Mathematics Subject Classification: 58-00, 58J00.Keywords and Phrases: Harmonic functions, Complete manifolds, Green’s func-

tions, 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 Lq Harmonic FunctionsReferences

0 IntroductionThe 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 ona complete manifold with nonnegative Ricci curvature, the issue of understandingvarious spaces of harmonic functions has been one of the central questions ingeometric analysis. During the last 20 years, there had been many significant

Research partially supported by NSF grant DMS-0503735.Department of Mathematics, University of California, Irvine CA 92697-3875, USA.E-mail: [email protected]

190 Peter Li

discoveries, but more importantly, the techniques developed in these theories areextremely useful when applied to other problems in geometic analysis. The point ofview taken in these notes is to describe some of the tools and techniques developedin the context of harmonic functions. The primary goal is to present the subjectto 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 treatmentto 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 EstimatesWe begin with a discussion on an important estimate that is essential to the

study of harmonic functions. In 1975, Yau [Y1] developed a maximum principlemethod to prove that complete manifolds with nonnegative Ricci curvature musthave a Liouville property. His argument was later localized in his joint paper withCheng [CY] and resulted in a gradient estimate for a rather general class of ellipticequations. In 1979 [L1], the maximum principle method was used by Li in provingeigenvalue estimates for compact manifolds. This method was then refined andused 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 versionof the gradient estimate for the parabolic Schrodinger equation. This philosophyhas 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-linearparabolic equations. In a recent work [LW9] of Li and Wang, they realized that onecan improve the gradient estimate of Yau and yield a sharp version of the estimatein which equality is achieved on a manifold with negative Ricci curvature. Thesharpness of this was somewhat unexpected since the parabolic gradient estimateis sharp on manifolds with nonnegative Ricci curvature.

In what follows, we will present the sharp gradient estimate of [LW9] forpositive functions satisfying the equation

∆f = −λ f

where λ ≥ 0 is a constant. Both the local and the global versions are included andvarious immediate consequences of the gradient estimate will also be stated.

Theorem 1.1 (Yau, Li-Wang). Let Mm be a complete manifold with Riccicurvature bounded from below by

RicM ≥ −(m− 1)K

for some constant K ≥ 0. If f is a positive function defined on the geodesic ballBp(2R) ⊂M satisfying

∆f = −λ f

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

f2(x) ≤ (4(m− 1)2 + 2ε)K

4− 2ε+ C((1 + ε−1)R−2 + λ),

for all x ∈ Bp(R) and for any ε < 2.Moreover, if f is defined on M , then

|∇f |2

f2(x) ≤ (m− 1)2K

2− λ+

√(m− 1)4K2

4− (m− 1)2λK.

and

λ ≤ (m− 1)2K

4.

Integrating this estimate along a geodesic joining two points in Bp(R2 ), onederives a Harnack type inequality.

Corollary 1.2. Let Mm be a complete manifold with Ricci curvature boundedfrom below by


for some constant K ≥ 0. If f is a positive function defined on the geodesic ballBp(2R) ⊂M satisfying

∆f = −λ f

for some constant λ ≥ 0, then there exists constants C9, C10 > 0 depending on m

such thatf(x) ≤ f(y)C9 exp(C10R

√K + λ)

for all x, y ∈ Bp(R2 ).

Theorem 1.1 also allows us to recover the upper bound of Cheng [Cg1] onthe bottom of the spectrum.

Corollary 1.3 (Cheng). Let Mm be a complete manifold with Ricci curvaturesatisfying

RicM ≥ −(m− 1)K,

for some constant K ≥ 0. If we denote λ1(M) to be the infimum of the spectrumof the Laplacian acting on L2 functions, then

λ1(M) ≤ (m− 1)2

4.

The Liouville theorems of Yau [Y1] and Cheng [Cg2] also follows as conse-quences.

192 Peter Li

Corollary 1.4 (Yau). Let Mm be a complete manifold with nonnegative Riccicurvature, then it does not admit any nonconstant, positive, harmonic functions.

Corollary 1.5 (Cheng). Let Mm be a complete manifold with nonnegative Riccicurvature. There exists a constant C(m) > 0, such that, for any harmonic functiondefined on M if we denote

s(R) = supx∈Bp(R)

|f(x)|

thensup

x∈Bp(R)

|∇f |(x) ≤ C R−1 s(2R).

In particular, M does not admit any nonconstant harmonic function satisfying thegrowth estimate

r−1(x) |f(x)| → 0

as x→∞.

To illustrate the sharpness of Theorem 1.1, we consider the following exam-ples.Example 1.6. Let Mm = R×Nm−1 be the complete manifold with the warpedproduct metric

ds2M = dt2 + exp(2t) ds2

N .

A direct computation shows that the Ricci curvature on M is given by

Ric1j = −(m− 1)δ1j , for 1 ≤ j ≤ m,

andRicαβ = exp(−2t) Ricαβ − (m− 1)δαβ , for 2 ≤ α, β ≤ m.

Here Ricαβ is the Ricci tensor on N and e1 = ∂∂t . In particular, if the Ricci

curvature of N is non-negative, then

RicM ≥ −(m− 1).

Moreover, N is Ricci flat if and only if M is Einstein with

RicM = −(m− 1).

Let f = exp(−αt) be a positive function defined on M for a constant m−12 ≤ α ≤

m− 1. One computes that

∆f =d2f

dt2+ (m− 1)

df

dt

= α2 f − (m− 1)α f

= −α(m− 1− α)f


and|∇f |2 = α2 f2.

When compare with the estimate of Theorem 1.1 usingK = 1 and λ = α(m−1−α),we observe that the inequality is in fact equality.Example 1.7 Let Mm = Hm be the hyperbolic m-space with constant curvature−1. Using the upper half-space model, Hm is given by Rm+ = (x1, x2, . . . , xm) |xm >

0 with metricds2 = x−2

m (dx21 + · · ·+ dx2

m).

For x = (x1, . . . , xm) let us consider the function f(x) = xm−1m . Direct computa-

tion yields

|∇f |2 = x2m

m∑i=1

(∂f

∂xi

)2

= (m− 1)2 x2m−2m

= (m− 1)2 f2.

and

∆f = x2m

m∑i=1

∂2f

∂x2i

− (m− 2)xm∂f

∂xm

= 0.

This implies that the gradient estimate for harmonic functions is sharp on Hm..

2 Green’s Function and ParabolicityLet Mm 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) definedon 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 satisfyDirichlet boundary condition, after integration by parts, (2.1) becomes∫

M

∆yG(x, y) f(y) dy = −f(x) (2.2)

194 Peter Li

which is equivalent to saying that

∆yG(x, y) = −δx(y), (2.3)

where δx(y) is the delta function at x. If we let f(x) = G(z, x), then (2.2) yields

G(z, x) = −∫M

∆yG(x, y)G(z, y) dy.

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 alsoobserve that by letting f(x) = ∆xG(z, x) in (2.2)

∆xG(z, x) = −∫M

∆yG(x, y) ∆yG(z, y) dy.

Since the right hand side is symmetric in x and z, we conclude that

∆xG(z, x) = ∆zG(x, z).

On the other hand,∆zG(x, z) = ∆zG(z, x)

by the symmetry of G, we conclude that

∆xG(z, x) = ∆zG(z, x).

Therefore (2.2) can be written as

∆x

∫M

G(x, y) f(y) dy =∫M

∆xG(x, y) f(y) dy

= −f(x) (2.5)

Hence if we define the operator ∆−1 by

∆−1f(x) = −∫M

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

then (2.1) and (2.5) gives∆ ∆−1 = I

and∆−1 ∆ = I,

respectively.


WhenMm is a complete, non-compact manifold without boundary, one wouldlike 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 ∈ C∞c (M).

For the case M = Rm, it is well-known that the function

G(x, y) =

(m(m− 2)ωm)−1 r(x, y)2−m for m ≥ 3

−(2ω1)−1 log(r(x, y)) for m = 2,

with ωm being the volume of the unit m-ball in Rm, will satisfy these properties.In fact, the following example will show that these formulas, when interpretedappropriately, reflect the ideal situations for manifolds also.Example 2.1. Let us now assume that Mm has a point p around which themetric is rotationally symmetric. This is equivalent to saying that if we take polarcoordinates (r, θ2, . . . , θm) around p, then

ds2M = dr2 + gαβ dθα dθβ

with gαβ(r) being functions of r alone. The Laplacian with respect to this coordi-nate system will take the form

∆ =∂2

∂r2+

1√g

∂√g

∂r

∂

∂r+ ∆∂Bp(r)

where g = det(gαβ) and ∆∂Bp(r) denotes the Laplacian defined on the sphere,∂Bp(r), of radius r centered at p. If we let Ap(r) to be the area of ∂Bp(r), then∫

√g dθ = Ap(r).

Since√g is independent of the θα’s,

1√g

∂√g

∂r=A′p(r)Ap(r)

,

and we have

∆ =∂2

∂r2+A′p(r)Ap(r)

∂

∂r+ ∆∂Bp(r).

Let us now consider the function

G(y) = −∫ r(y)

1

dt

Ap(t)

where r(y) is the distance from p to y. Direct computation gives

∆G =∂2G

∂r2+A′p(r)Ap(r)

∂G

∂r

= 0

196 Peter Li

for y 6= p. Moreover, if f ∈ C∞c (M) is a compactly supported smooth function,then ∫

M\Bp(ε)

G(y) ∆f(y) dy =∫M\Bp(ε)

∆G(y) f(y) dy

−∫∂Bp(ε)

G(y)∂f

∂rdθ +

∫∂Bp(ε)

∂G

∂rf dθ

=∫ ε

1

dt

Ap(t)

∫∂Bp(ε)

∂f

∂rdθ − 1

Ap(ε)

∫∂Bp(ε)

f dθ.(2.6)

Note that the continuity of f implies that

1Ap(ε)

∫∂Bp(ε)

f dθ → f(p)

as ε→ 0. Also the continuity of ∆f and the divergence theorem imply that

1Vp(ε)

∫∂Bp(ε)

∂f

∂rdθ =

1Vp(ε)

∫Bp(ε)

∆f dy

→ ∆f(p)

as ε→ 0, where Vp(r) is the volume of Bp(r). This implies that∫∂Bp(ε)

∂f

∂rdθ ∼ ∆f(p)Vp(ε).

On the other hand, sinceAp(t) ∼ mωm t

m−1,

we have ∫ ε

1

dt

Ap(t)∼

− 1m(m− 2)ωm

ε2−m for m ≥ 3

12ω2

log(ε) for m = 2.

Therefore, by letting ε→ 0, the right hand side of (2.6) becomes

limε→0

(∫ ε

1

dt

Ap(t)

∫∂Bp(ε)

∂f

∂rdθ − 1

Ap(ε)

∫∂Bp(ε)

f dθ

)= −f(p),

hence ∫M

G(y) ∆f(y) dy = −f(p).

In particular, this implies that one can take

G(p, y) = −∫ r(y)

1

dt

Ap(t)(2.7)


to be a Green’s function. Note that if∫ ∞1

dt

Ap(t)<∞

then by adding this constant to G, we can use the formula

G(p, y) =∫ ∞r(y)

dt

Ap(t). (2.8)

It is in the form of (2.7) and (2.8) that the Green’s function in Rm is given by forthe cases m = 2 and m ≥ 3, respectively.

The existence of a Green’s function on a general complete manifold was firstproved by Malgrange [M]. However, for the purpose of application, a constructiveargument is a key step in getting the appropriate estimates. The first constructiveproof was published in [LT2], and we will give a brief description below.

Let Ωi∞i=1 be a compact exhaustion of M , such that

Ωi ( Ωj for i < j,

∪∞i=1Ωi = M,

and each Ωi is a sufficiently smooth compact subdomain of M. For each i, letGi(x, y) be the symmetric, positive, Green’s function with Dirichlet boundarycondition of Ωi. Moreover,

Gi(p, y) ∼

(m(m− 2)ωm)−1 r2−m(p, y) for m ≥ 3

−(2ω2)−1 log r(p, y) for m = 2,

as y → p. The construction argument of [LT2] asserts that if Mm admits a positiveGreen’s function, G(p, y), then G can be obtained by

G(p, y) = limi→∞

Gi(p, y).

Theorem 2.1. Let Mm be a complete manifold without boundary. There existsa 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 ifthere exists a positive superharmonic function f on M \ Bp(R) with the propertythat

lim infx→∞

f(x) < infx∈∂Bp(R)

f(x).

The existence and nonexistence of a positive Green’s function divides the theclass of complete manifolds into two categories. In general the methods in dealingwith function theory on these manifolds are different. Hence it is important tounderstand the difference in the two categories.

198 Peter Li

Definition 2.2. A complete manifold is said to be parabolic if it does not admita 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 thereexists 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 un-bounded connected component of M \ Ω. The number of ends with respect of Ω,denoted by NΩ(M), is the number of unbounded connected component of M \ Ω.

It is obvious that if Ω1 ⊂ Ω2, then NΩ1(M) ≤ NΩ2(M). Hence if Ωiis a compact exhaustion of M , then NΩi(M) is a monotonically nondecreasingsequence. 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) = maxi→∞

NΩi(M).

One readily checks that this is independent of the compact exhaustion Ωi. Infact, it is also easy to see that there must be an i0 such that N(M) = NΩi(M),for all i ≥ i0. Hence for all practical purposes, we may assume that M \ Bp(R0)has N(M) number of unbounded connect components, for some R0.

In general, when we say that E is an end we mean that it is an end withrespect to some compact subset Ω. In particular, its boundary ∂E is given by∂Ω ∩ E.

Definition 2.4. An end E is said to be parabolic if it does not admit a positiveharmonic function f satisfying

f = 1 on ∂E

andlim infy→E(∞)

f(y) < 1,

where E(∞) denotes the infinity of E. Otherwise, E is said to be nonparabolicand 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 N0(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 assumethat the function f in Definition 2.4 satisfied

f |∂E = 1 and lim infy→E(∞)

f(y) = 0. (2.9)


Also note that if E is a nonparabolic end of M , then by extending f to be identi-cally 1 on (M \Ω) \E, it can be used to construct a positive Green’s function onM. 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 alsopoint out that E being nonparabolic is equivalent in saying that E has a positiveGreen’s function with Neumann boundary condition.

If E is parabolic, one can also construct a barrier function g which is harmonicon E and have the properties that

g|∂E = 0 and supy∈E

g =∞. (2.10)

In the case when M is nonparabolic, it follows that there is a unique minimalpositive Green’s function. The following estimate for the minimal Green’s functionof 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 nonparabolicthen for any point p ∈M ∫ ∞

1

dt

Ap(t)<∞, (2.11)

where Ap(r) denotes the area of ∂Bp(r). Moreover, if G(p, y) is the minimalGreen’s function, then∫ r

1

dt

Ap(t)≤ supy∈∂Bp(1)

G(p, y)− infy∈∂Bp(r)

G(p, y)

for all r > 1.

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 showthat (2.11) indeed implies nonparabolicity. We will refer the readers to [LT7] foradditional 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 Riemannianmanifold. Suppose for some ν ≥ 1 and C > 0, E satisfies a Sobolev type inequalityof the form (∫

E

|u|2ν) 1ν

≤ C∫E

|∇u|2

for all compactly supported function u ∈ H1,2c (E) defined on E, then E must either

have finite volume or be nonparabolic.

On the other hand, we also have the following proposition.

200 Peter Li

Proposition 2.8. Let M be a complete Riemannian manifold. Suppose for someν > 1 and C > 0, M satisfies a Sobolev type inequality of the form(∫

E

|u|2ν) 1ν

≤ C∫E

|∇u|2

for all compactly supported function u ∈ H1,2c (M). There exists a constant C1 > 0

depending only on C and ν, such that, the volume of a geodesic ball of radius Rcentered at p ∈M must satisfy

Vp(R) ≥ C1R2νν−1 .

In particular, each end of M must have at least R2νν−1 volume growth.

Combining Propositions 2.7 and 2.8, we obtain the following corollary for theSobolev type inequality with ν > 1. The case when ν = 1 is just the DirichletPoincare inequality and in that case, it is possible to have a finite volume endgiven by a cusp.

Corollary 2.9. Let E be an end of a complete Riemannian manifold. Supposefor some ν > 1 and C > 0, E satisfies a Sobolev type inequality of the form(∫

E

|u|2ν) 1ν

≤ C∫E

|∇u|2

for all compactly supported function u ∈ H1,2c (E) defined on E, then E must be

nonparabolic.

3 Heat Kernel Estimates and Mean ValueInequality

In this section, we will present the estimates for positive solutions of theheat equation proved by Li and Yau in [LY2]. In particular, upper bound for thefundamental solution of the heat equation will be established for manifolds withRicci curvature bounded from below.

The gradient estimate given in Theorem 3.1 has fundamental importance instudying parabolic equations. There are many subsequent development of similartype estimates for other nonlinear partial differential equations. However, we willnot discuss these directions since they are out of the scope of these notes. Thepurpose of this section is to present the essential estimates that are necessary forour purpose and for the proof of the the mean value inequality.

Theorem 3.1 (Li-Yau). Let Mm be a complete manifold with boundary. Assumethat p ∈ M and R > 0 so that the geodesic ball Bp(4R) does not intersect theboundary of M. Suppose the Ricci curvature of M on Bp(4R) is bounded frombelow by



for some constant K ≥ 0. If f(x, t) is a positive solution of the equation(∆− ∂

∂t

)f(x, t) = 0

on M × [0, T ], then for any 1 < α ≤ 2, the function h(x, t) = log f(x, t) satisfiesthe estimate

|∇h|2 − αht ≤m

2α2t−1 + C1 (α− 1)−1 (R−2 +K)

on Bp(2R)× t for 0 ≤ t ≤ T , where C1 is a constant depending only on m.

This estimate takes a much simpler form when the function is defined globallyon a manifold with nonnegative Ricci curvature. In fact, the inequality is sharpon Rm.

Corollary 3.2. Let Mm be a complete manifold with nonnegative Ricci curvature.If f(x, t) is a positive solution of the heat equation(

∆− ∂

∂t

)f(x, t) = 0

on M × [0,∞), then|∇f |2

f2− ftf≤ m

2t

on M × [0,∞).

Theorem 3.1 implies a Harnack type inequality, which is extremely useful inthe study of parabolic equations. One simply integrate the gradient estimate alonga suitably chosen curve in M × R joining the points (x, t1) and (y, t2).

Corollary 3.3. Under the same hypotheses of Theorem 3.1,

f(x, t1) ≤ f(y, t2)(t2t1

)mα2

exp(α r2(x, y)4(t2 − t1)

+ C1(α− 1)−1(R−2 +K) ((t2 − t1))

for any x, y ∈ Bp(R) and 0 < t1 < t2 ≤ T, where r(x, y) is the geodesic distancebetween x and y.

Using Theorem 3.1, we can derive an upper bound on the fundamental solu-tion of the heat equation.

Theorem 3.4 (Li-Yau). Let Mm be a complete manifold and H(x, y, t) denotesthe minimal symmetric heat kernel defined on M ×M × (0,∞) with the propertiesthat (

∆y −∂

∂t

)H(x, y, t) = 0

202 Peter Li

andlimt→0

H(x, y, t) = δx(y).

For any p ∈M , ε > 0, R > 0 and t ≤ R2

4 , if the Ricci curvature of M on Bp(4R)is bounded from below by


then there exists constants C1 > 0 depending only on m and ε and C2 > 0 depend-ing only on m, such that,

H(x, y, t) ≤ C1 exp(−λ1 t)V− 1

2x (√t)V −

12

y (√t)

× exp(− r2(x, y)

4(1 + 2ε)t+√C2(R−2 +K)t

)for any x, y ∈ Bp(R) and t ≤ R2

4 .

Another important inequality is the mean value inequality for positive sub-harmonic functions. The classical mean value property for harmonic functions inRm asserts that the average value of a harmonic function on a ball is given by thevalue of the function at the center point, i.e.,

V −1p (R)

∫Bp(R)

f = f(p).

If f is a nonnegative subharmonic function on Rm, it must satisfy the inequality

V −1p (R)

∫Bp(R)

f ≥ f(p).

The validity of such inequality on a manifold is important in the study of nonlinearanalysis and certainly for the study of harmonic functions. Of course, the preciseform of the inequality might not be the same as the one in Rm, but one expects amean value inequality of the form

C V −1p (R)

∫Bp(R)

f ≥ f(p),

where C > 0 is a constant that may depend on M. By using the Sobolev inequalityand Moser’s iteration method, one can obtain the mean value inequality in theabove form. However, the constant C will depend on the Sobolev constant. Onecan then estimate the Sobolev constant by estimating the isoperimetric inequalityas in [Cr] (also see [L4]), but the estimate will depend on the volume Vp(R) andother geometric quantities. On the other hand, it is often desirable that theconstant C is independent of the volume so that the left hand side is indeedthe average value of f . In [LS], Li and Schoen proved a mean value inequalitywhere the constant C only depends on the lower bound of the Ricci curvature, m


and R. Their argument works for nonnegative functions satisfying the differentialinequality

∆f ≥ −λ f (3.1)

for 0 ≤ λ ≤ λ1(Bp(R)) where λ1(Bp(R)) is the first Dirichlet eigenvalue of Bp(R).In other applications, it is useful to derive a mean value inequality for nonnegativefunction f satisfying (3.1) without any restriction on λ. This was done by Li andTam [LT5] using the upper bound of the heat kernel. They actually proved a meanvalue inequality for nonnegative subsolutions of the heat equation, which includessubsolutions of (3.1) as a special case. We would like to point out that the value ofthe mean value constant is also of significance. One may refer to [LW2] for furtherdetails.

Theorem 3.5 (Li-Tam). Let Mm be a complete noncompact Riemannian man-ifold with boundary. Let p ∈ M and R > 0 be such that the geodesic ball Bp(2R)does not intersect the boundary of M. Suppose g(x, t) is a nonnegative functiondefined on Bp(2R)× [0, T ] for some 0 < T ≤ R2

4 satisfying the differential inequal-ity

∆g − ∂g

∂t≥ 0.

If the Ricci curvature of Bp(2R) is bounded by


for some constant K ≥ 0, then for any q > 0, there exists positive constants C1

and C2 depending only on m and q such that for any 0 < τ < T , 0 < δ < 12 , and

0 < η < 12 ,

supBp((1−δ)R)×[τ,T ]

gq ≤ C1V (K, 2R)Vp(R)

(R√K + 1

)exp(C2

√K T )

×(

1δR

+1√ητ

)m+2 ∫ T

(1−η)τ

ds

∫Bp(R)

gq(y, s) dy,

where V (K, r) is the volume of the geodesic ball of radius r in m-dimensionalconstant curvature space form with constant sectional curvature −K.

Corollary 3.6 (Li-Schoen, Li-Tam). Let Mm be a complete manifold withboundary and p be a fixed point in M. Suppose R > 0 be such that Bp(2R) doesnot intersect the boundary of M and assume that the Ricci curvature of M onBp(2R) satisfies the bound


for some constant K ≥ 0. Let 0 < δ < 12 , q > 0, and λ ≥ 0 be fixed constants. Then

there exists a constant C > 0 depending only on δ, q, λR2, m, and R√K, such

204 Peter Li

that, for any nonnegative function f defined on Bp(2R) satisfying the differentialinequality

∆f ≥ −λ f,

we havesup

Bp((1−δ)R)

fq ≤ C V −1p (R)

∫Bp(R)

fq(y) dy.

4 Harmonic Functions and EndsA 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. Theconstruction was first proved by Tam and Li in [LT1] for manifolds with nonnega-tive sectional curvature near infinity. They later gave a construction for arbitrarycomplete manifolds in [LT6]. In [STW], Sung, Tam, and Wang presented theconstruction in a more systematic manner and gave general criteria on the extend-ability of a harmonic function that is defined only near infinity. In particular, theseharmonic 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 K0(M) and K′(M), with (possibly infi-nite) dimensions given by k0(M) and k′(M), respectively, such that

k0(M) = N0(M)

andk′(M) = N ′(M).

In particular,k0(M) + k′(M) = N(M).

Moreover, K0(M) is a subspace of the space of bounded harmonic functions withfinite Dirichlet integral on M, and K′(M) is spanned by a set of positive harmonicfunctions.

A similar construction also gives a corresponding theorem for parabolic man-ifolds.

Theorem 4.2. Let M be a complete manifold that is parabolic. There exists aspace of harmonic functions K(M), with (possibly infinite) dimension given byk(M), such that

k(M) = N(M),

Moreover, K(M) is spanned by a set of harmonic functions which are boundedeither 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 ofa complete manifold whose Ricci curvature is “almost” nonnegative. In particular,


we will give an estimate on the number of ends of such a manifold. When M hasnonnegative Ricci curvature everywhere, this argument also recovers the splittingtheorem of Cheeger and Gromoll [CG]. It was proved by Li and Tam [LT6] that ifwe assume that the Ricci curvature of M is bounded from below by

RicM (x) ≥ −(m− 1)k(r(x))

for some nonincreasing function k(r) ≥ 0 satisfying the property∫ ∞1

k(r) rm−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 beused in many of the theorems stated in these notes.

Lemma 4.3 (Li). Let Ω ⊂ Mm 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 ofsections of T . Then there exists a section f ∈ F such that

dimF∫

Ω

|f |2 ≤ nV (Ω) supΩ|f |2,

where dimF is the dimension of F and |f | is the norm with respect to the innerproduct of E.

Theorem 4.4 (Li-Tam). Let Mm be a complete manifold and p ∈M be a fixedpoint such that RicM (x) ≥ −k(r(x)) where r(x) denotes the distance to the pointp. Suppose that k : [0,∞) → [0,∞) is a nonincreasing continuous function suchthat

∫∞0rm−1k(r)dr < ∞. Then there exists a constant C(m, k) depending only

on m and k, such that, the number of ends of M is bounded by

N(M) ≤ C(m, k).

Corollary 4.5. Let Mm be a complete manifold. Suppose Ω ⊂ Bp(R0) is acompact 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, R0 and the lower boundof the Ricci curvature on Bp(R0), such that, M has at most C number of ends.

Moreover, if M has nonnegative Ricci curvature everywhere then M eitherhas only one end, or it must be isometrically a cylinder M = R×N given by theproduct metric, where N is a compact manifold with nonnegative Ricci curvature.

206 Peter Li

5 Stability of Minimal HypersurfacesIn this section, we will give applications of harmonic functions to the study

of complete minimal hypersurfaces. Let Nm+1 be a complete manifold with non-negative Ricci curvature. Suppose Mm is a complete minimal hypersurface in N .If |A|2 denotes the square of the length of the second fundamental form of M andRicN (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

ψ2 |A|2 +∫M

ψ2 RicN (ν, ν) ≤∫M

|∇ψ|2 (5.1)

for any compactly supported function ψ ∈ H1,2c (M). Geometrically the stability

inequality is derived from the second variation formula for the volume functionalunder normal variations. Hence a stable minimal hypersurface is not only a crit-ical point of the volume functional but its second derivative is nonnegative withrespect to any normal variations. The elliptic operator associated to the stabilityinequality is given by

L = ∆ + |A|2 + RicN (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 manynegative eigenvalues. This has the geometric interpretation that there is only afinite dimensional space of normal variations violating the stability inequality.

The study of stable minimal hypersurfaces can be viewed as an effort ofproving a generalized Bernstein’s theorem. Bernstein first established that anentire minimal graph - a minimal hypersurface which is given by a graph of afunction defined on R2. - in R3 must be a plane. The validity of Bernstein theoremin higher dimension was established for entire minimal graphs in Rm+1 for m ≤ 7by Simons [S], and many other authors, such as Fleming [Fl], Almgren [A], andDeGiorgi [De], for the lower dimensional cases. Counter-examples for m ≥ 8 wasfound by Bombieri, DeGiorgi, and Guisti [BDG]. Since entire minimal graphs arearea minimizing, a natural question is to ask if a Bernstein type theorem is validfor stable minimal hypersurfaces in Rm+1.

In 1979, do Carmo and Peng [dCP] proved that a complete, stable, mini-mally immersed hypersurface M in R3 must be planar. At the same time, Fischer-Colbrie and Schoen [FS] independently showed that a complete, stable, minimallyimmersed hypersurface M in a complete 3-dimensional manifold N with nonneg-ative scalar curvature must be either conformally a plane R2 or conformally acylinder R×S1. For the special case when N is R3, they also proved that M mustbe planar.

In 1984, Gulliver [G1] studied a yet larger class of submanifolds in R3. Heproved that a complete, oriented, minimally immersed hypersurface with finiteindex in R3 must have finite total curvature. In particular, after applying Huber’stheorem, one concludes that the hypersurface must be conformally equivalent to


a compact Riemann surface with finitely many punctures. The same result wasalso independently proved by Fischer-Colbrie in [F]. In addition, she also provedthat a complete, oriented, minimally immersed hypersurface with finite index in acomplete 3-dimensional manifold with nonnegative scalar curvature must be con-formally equivalent to a compact Riemann surface with finite punctures. Shortlyafter, Gulliver [G2] improved the result of Fischer-Colbrie and showed that if theambient manifold has nonnegative scalar curvature then a minimal hypersurfacewith finite index must have quadratic area growth, finite topological type, andthe length of the second fundamental form must be square integrable. Indeed, acomplete surface with quadratic area growth and finite topological type must beconformally equivalent to a compact Riemann surface with finitely many punc-tures.

In 1997, Cao, Shen, and Zhu [CSZ] considered the high dimensional cases ofthe theorem of do Carmo-Peng and Fischer-Colbrie-Schoen. They proved that acomplete, oriented, stable, minimally immersed hypersurface Mn in Rn+1 musthave only one end. This theorem was generalized by Li and Wang [LW6] (seeTheorem 5.6), where they showed that a complete, oriented, minimally immersedhypersurface Mn in Rn+1 with finite index must have finitely many ends. In a laterpaper [LW8], Li and Wang also generalized their theorem to minimal hypersurfaceswith finite index in a complete manifold with nonnegative sectional curvature.

In this section, we will first present the relationship between harmonic func-tions and stability of minimal hypersurfaces. Taken the point of view of functiontheory, one readily recovers most of the two dimensional results stated earlier. Wealso present higher dimensional results that can be obtained using this point ofview.

Note that when the ambient manifold is Euclidean space and n ≥ 3, thenM is non-parabolic by applying Proposition 2.8. The key issue is the validity ofSobolev inequality proved by Michael and Simon [MS] in the form(∫

M

|u|2mm−2

)m−2m

≤ C∫M

|∇u|2

on any minimal submanifolds of Rn. However, when N is only assumed to havenonnegative Ricci (or sectional) curvature, then M can be parabolic as in the caseof the cylinder M = R×P in N = R2 ×P. The next theorem states that the caseM being parabolic is a very special situation.

Proposition 5.1. Let Mm be a complete, minimally immersed, stable, hypersur-face in a manifold, Nm+1, with nonnegative Ricci curvature. If M is parabolic,then it must be totally geodesic in N . Moreover, the Ricci curvature RicN (ν, ν)of N in the normal direction to M also vanishes, and M must have nonnegativescalar curvature.

Proposition 5.1 reduces our study of stable minimal hypersurfaces to thenonparabolic case. The following lemma of Schoen and Yau [SY] illustrates therole of harmonic functions in the stability inequality.

208 Peter Li

Lemma 5.2 (Schoen-Yau). Let Mm be a complete, minimally immersed, stable,hypersurface in Nm+1. Suppose N has nonnegative sectional curvature and u is aharmonic function defined on M . Then the inequality

1m

∫M

φ2 |A|2 |∇u|2 +1

m− 1

∫M

φ2 |∇|∇u||2 ≤∫M

|∇φ|2 |∇u|2

holds for any compactly supported, nonnegative function φ ∈ H1,2c (M).

In the case when m = 2, Schoen and Yau [SY] observed that the stabilityinequality can be rewritten using the scalar curvature of N , denoted by SN . Onealso observed that the stability inequality can be lifted to the universal covering ofM , and as a corollary of Proposition 5.1 and Lemma 5.2, one readily recovers thetheorems of Fischer-Colbrie and Schoen [FS] (also by do Carmo and Peng [dCP]for the special case when N = R3).

Theorem 5.3 (Fischer-Colbrie-Schoen, do Carmo-Peng). Let M2 be anoriented, complete, stable, minimal hypersurface in a complete manifold N3 withnonnegative scalar curvature. Then M must be conformally equivalent to eitherthe complex plane C or the cylinder R× S1. If M is conformally equivalent to thecylinder and has finite total curvature, then it must be totally geodesic and thescalar curvature of N along M must be identically 0.

Corollary 5.4 (Fischer-Colbrie-Schoen). Let M2 be an oriented, complete,stable, minimal hypersurface in a complete manifold N3 with nonnegative Riccicurvature. Then M must be totally geodesic in N and the Ricci curvature of Nin the normal direction to M must be identically zero along M. Moreover, M iseither

(1) conformally equivalent to the complex plane C; or(2) isometrically the cylinder R× S1.

Moreover, if N = R3, then M must be planar in R3.

Let us now consider the case when m > 2. We assume that Nm+1 is acomplete manifold with nonnegative sectional curvature.

Theorem 5.5 (Li-Wang). Let Mm be a complete, stable, minimally immersedhypersurface in Nm+1. Suppose N is a complete manifold with nonnegative sec-tional curvature. If M is parabolic, then it must be totally geodesic and has nonneg-ative sectional curvature. In particular, M either has only one end, or M = R×Pwith 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 Mm be a complete, properly immersed, stable, minimal hy-persurface in Nm+1. Suppose N is a complete manifold with nonnegative sectionalcurvature. Then either

(1) M has only one end; or


(2) M = R×P with the product metric, where P is compact with nonnegativesectional curvature, and M is totally geodesic in N .

This corollary includes the theorem of Cao, Shen, and Zhu [CSZ], where theygave the first structural description of a stable minimal hypersurface in Rm+1 form ≥ 3. Using harmonic functions, the number of ends can be seen to be boundedfor the minimal hypersurface has finite index.

Theorem 5.7 (Li-Wang). Let Mm be a minimally immersed hypersurface inRm+1 with m ≥ 3. If M has finite index then there exist a constant C > 0 depend-ing on M such that

N(M) ≤ C.

6 Polynomial Growth Harmonic FunctionsIn this section, we will consider polynomial growth harmonic functions on a

complete manifold. Recall that any polynomial growth harmonic function in Rm isnecessarily a polynomial with respect to the variables in rectangular coordintates.Hence the space of polynomial growth harmonic functions of at most order d aregiven 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

Hd(M) = f |∆f = 0, and |f(x)| ≤ C rd(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 Hd(M) byhd(M).

Using this notation, one computes that

hd(Rm) =(m+ d− 1

d

)+(m+ d− 2d− 1

).

On the other hand, Cheng’s theorem (Corollary 1.5) asserts that if M has non-negative Ricci curvature, then

hd(M) = 1

for all d < 1. In view of this, Yau conjectured ([Y3] problem 48) that hd(M)must be finite dimensional if M has nonnegative Ricci curvature. He also raised asimilar questions for holomorphic sections of vector bundles. In 1989, Li and Tam[LT3] proved that

h1(M) ≤ m+ 1

if M has nonnegative Ricci curvature. Note that this estimate is sharp and it isachieved by Rm.

210 Peter Li

Theorem 6.2 (Li-Tam). Let M be a complete manifold with nonnegative Riccicurvature. Suppose the volume growth of M satisfies

lim supR→∞

R−n Vp(R) <∞

for some n ≤ m, thenh1(M) ≤ n+ 1 ≤ m+ 1.

In fact, Li [L5] showed that if M is Kahler and has nonnegative holomorphicbisectional curvature then equality holds if and on if M = Cn with 2n = m. Thereal case of this theorem was later proved by Cheeger, Colding, and Minicozzi[CCM]. They showed that if M has nonnegative Ricci curvature and h1(M) =m+ 1, then M = Rm.

In the case of Yau’s conjecture for polynomial growth harmonic functions, itwas first proved by Colding and Minicozzi [CM1-3] and they gave the estimate

hd(M) ≤ C dm−1

for some constant C > 0 depending on M . In [L6], Li gave a simplified proof ofthis estimate that holds for a much larger class of manifolds, which we will nowgive a brief description.

Definition 6.3. A manifold is said to satisfy a volume comparison condition (Vµ)for some µ > 1, if for any point x ∈ M and any real numbers 0 < R1 ≤ R2 < ∞the volume of the geodesic balls centered at x must satisfy

Vx(R2) ≤ Vx(R1)(R2

R1

)µ.

Definition 6.4. A manifold is said to satisfy a mean value inequality (M) if thereexists a constant C0 > 0 such that for any x ∈M and R > 0, the inequality

f2(x) ≤ C0 V−1x (R)

∫Bx(R)

f2(y) dy

is valid for all nonnegative subharmonic function f.

Note that if M has nonnegative Ricci curvature then M satisfies condition(Vm) by the Bishop volume comparison theorem, and condition (M), by the the-orem of Li-Schoen [LS] (Corollary 3.6).

Theorem 6.5 (Li). Let Mm be a complete manifold satisfying conditions (Vµ)and (M). Suppose E is a rank-n vector bundle over M. Let Sd(M,E) ⊂ Γ(E) bea linear subspace of sections of E, such that, all u ∈ Sd(M,E) satisfy

(a) ∆|u| ≥ 0, and(b) |u|(x) ≤ O(rd(x)) as x→∞.


Then the dimension of Sd(M,E) is finite. Moreover, for all d ≥ 1, there exists aconstant C > 0 depending only on µ such that

dimSd(M,E) ≤ nC C0 dµ−1.

In particular, we recover the theorem of Colding and Minicozzi and alsoanswered Yau’s question on sections of certain class of vector bundles.

Corollary 6.6 (Colding-Minicozzi). Let Mm be a complete Riemannian man-ifold with nonnegative Ricci curvature. There exists a constant C > 0 dependingonly on m, such that, dimension of the Hd(M) is bounded by

hd(M) ≤ C dm−1

for all d ≥ 1.

In fact, finite dimensionality of Hd(M) can be obtained by substantiallyrelaxing both the volume comparison condition and the mean value inequalitycondition. 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 valueinequality (WM) if there exists constants C0 > 0 and b > 1 such that, for anynonnegative subharmonic function f defined on M , it must satisfy

f(x) ≤ C0 V−1x (R)

∫Bx(bR)

f(y) dy.

for all x ∈M and R > 0.

Theorem 6.8 (Li). Let M be a complete manifold satisfying the weak mean valueproperty (WM). Suppose that the volume growth of M satisfies Vp(R) = O(Rµ)as R→∞ for some point p ∈M. Then Hd(M) is finite dimensional for all d ≥ 0and dimHd(M) ≤ C0 (2b+ 1)(2d+µ).

Definition 6.9. A complete manifold M is said to satisfy a Sobolev inequality(S) if there exist constants C1 > 0 and µ > 2, such that, for all p ∈ M, R > 0,and for all f ∈ Hc

1,2(M), we have(∫Bp(R)

|f |2µµ−2

)µ−2µ

≤ C1 Vp(R)−2µ

∫Bp(R)

(R2 |∇f |2 + f2).

Corollary 6.10. Let M be a complete manifold with property (S). Then

dimHd(M) < C βd

for all d ≥ 0 for some constant C1 > 0 and β > 1 depending only on C0, µ, and b.

If we assume that Mm has nonnegative sectional curvature then Li and Wang[LW3] proved that the quantity

∑di=1 h

i(M) has an upper bound that is asymp-totically sharp as d→∞.

212 Peter Li

Theorem 6.11 (Li-Wang). Let Mm be a complete manifold with nonnegativesectional curvature. Let 0 ≤ α0 ≤ ωm be a constant such that

lim infr→∞

r−m Vp(r) = α0.

If hd = dimHd(M) denotes the dimension of the space of polynomial growth har-monic functions of at most degree d, then

lim infd→∞

d−(m−1) hd ≤ 2α0

(m− 1)!ωm.

Moreover, the equality

lim infd→∞

d−(m−1) hd =2

(m− 1)!holds if and only if M = Rm.

Theorem 6.1 and Theorem 6.11 indicated that on a complete manifold withnonnegative sectional curvature hd(M) is bounded from above by hd(Rm) whend ≤ 1 or when d→∞. One conjectures that perhaps the inequality

hd(M) ≤ hd(Rm)

is valid for all d. It is also interesting to point out that this is not true whenthe assumption is relaxed to nonnegative Ricci curvature, as pointed out by anexample of Donnelly [D].

Similar studies on solutions of general elliptic operators on Rm can also befound in the works of Avellaneda-Lin [AL], Lin [Ln], Moser-Struwe [MrS], andLi-Wang [LW4-5].

7 Massive Sets and the Structure of HarmonicMaps

In this section, we will introduce the notion of d-massive sets. The notionof 0-massive set was first introduced by Grigor’yan [G] where he established therelationship between massive sets and harmonic functions. While it is not clear ifsuch a relationship exists for d > 0, but the notion of d-massive sets have importantgeometric and analytic implications.

Definition 7.1. For any real number d ≥ 0, a d-massive set Ω is a subset of amanifold M that admits a nonnegative, subharmonic function f defined on Ω withthe boundary condition

f = 0 on ∂Ω,

and satisfying the growth property

f(x) ≤ C rd(x)

for all x ∈ Ω and for some constant C > 0. The function f is called the potentialfunction of Ω. We also denote md(M) to be the maximum number of disjointd-massive sets admissible on M.

The following theorem was proved by Grigor’yan [G].


Theorem 7.2 (Grigor’yan). Let Mm be a complete Riemannian manifold. Themaximum number of disjoint 0-massive sets admissible on M is given by the di-mension of the space of bounded harmonic functions on M, i.e.,

m0(M) = h0(M).

As pointed out earlier, there are no direct relationship between hd(M) andmd(M) for d > 0. However, Li and Wang [LW1, LW3] discovered that there areparallel theories on the two numbers that provide a philosophical connection. Thisis demonstrated by the following theorem that mirrors Theorem 6.6 when appliedto polynomial growth harmonic functions.

Theorem 7.3 (Li-Wang). Let Mm be a complete manifold satisfying conditions(Vµ) and (M). For all d ≥ 1, there exists a constant C > 0 depending only on µ

such thatmd(M) ≤ C C0 d

µ−1.

Similarly, we also have the following finiteness theorem that mirrors Theorem6.9.

Theorem 7.4 (Li-Wang). Let M be a complete manifold satisfying the weakmean value property (WM). Suppose that the volume growth of M satisfiesVp(R) = O(Rµ) as R→∞ for some point p ∈M. Then

md(M) ≤ C0 (2b+ 1)(2d+µ).

One can also prove a sharp estimate when M = R2.

Theorem 7.5 (Li-Wang). On R2,

md(R2) ≤ 2d

for all d > 0.

We will now apply the notion of massive sets to study the structure of theimage of a harmonic map into a Cartan-Hadamard manifold. More specifically, Liand Wang [LW1] developed this theory that applies to the target being a stronglynegatively curved Cartan-Hadamard manifold, or when it is a 2-dimensional vis-ibility manifold. The connection between massive sets and harmonic maps intoCartan-Hadamard manifold follows from the fact that the pullback of a convexfunction by harmonic map is a subharmonic function. Since any Busemann func-tion on a Cartan-Hadamard manifold is convex, the supports of their pullback bya 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 sectionalcurvature. It is well known that N can be compactified by adding a sphere at

214 Peter Li

infinity S∞(N). The resulting compact space N = N ∪ S∞(N) is homeomorphicto a closed Euclidean ball. Two geodesic rays γ1 and γ2 in N are called equivalentif their Hausdorff distance is finite. Then the geometric boundary S∞(N) is simplygiven by the equivalence classes of geodesic rays in N . A sequence of points xiin N converges to x ∈ S∞(N) if for some fixed point p ∈ N , the sequence ofgeodesic rays pxi converges to a geodesic ray γ ∈ x. In this case, we say γ isthe geodesic segment px joining p to x. Recall that a subset C in N is strictlyconvex if any geodesic segment between any two points in C is also contained inC. For a subset K in N , the convex hull of K, denoted by C(K), is defined tobe the smallest strictly convex subset C in N containing K. The convex hull canalso be obtained by taking the intersection of all convex sets C ⊂ N containingK. When N is a Cartan-Hadamard manifold, there is only one geodesic segmentjoining 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 bya geodesic segment given by a geodesic line, as indicated by two non-antipodalpoints in S∞(Rn). If every pair of points in S∞(N) can be joined by a geodesicline in N , then N is said to be a visibility manifold. This class of manifolds wasextensively studied by Eberlein and O’Neill [EO]. A typical example of a visibilitymanifold is a Cartan-Hadamard manifold with sectional curvature bounded fromabove by a negative constant −a < 0.

To remedy the situation when N is not a visibility manifold, we defined ageneralized 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 γi in N with end pointsxi and yi such that xi → x and yi → y. We will denote γ by xy.

Observe that if xy ∩ S∞(N) = x, y, then xy must be a geodesic line inN and hence a geodesic segment in the traditional sense. For the case of twonon-antipodal points in S∞(R2), the shortest arc on S1 = S∞(R2) joining the twopoints will be the geodesic segment in the sense defined above. If the two pointsare antipodal in S∞(R2), say the northpole and the southpole, then there areinfinitely many geodesic segments joining them. Each vertical line is a geodesicsegment in the genuine sense. Also, both arcs on S1 joining the two poles aregeodesic segments joining them. Using this definition, for a pair of points inS∞(N), it is possible to have more than one geodesic segments joining them. Theconvexity 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.


Definition 7.8. For a subset A in N , we define its convex hull C(A) to be thesmallest convex subset of N containing A.

In what follows, when we say that a subset is closed, we mean that it is closedin N unless otherwise noted. In general, we denote the closure for a subset A inN by A. For a given sequence of closed subsets Ai decreasing to A, it is naturalto ask whether the convex hull of Ai in N decreases to the convex hull of A. Forthis purpose, we introduce the following definition.

Definition 7.9. A Cartan-Hadamard manifold N is said to satisfy the separationproperty if for every closed convex subset A in N and every point p not in A, thereexists 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 manifoldwith constant negative curvature, it is easy to check that the separation propertyholds. In fact, for a point p not in the closed convex set A, pick up a pointq ∈ A such that r(p, q) = r(p,A). Then the convexity of A and the first variationformula imply that for z ∈ A, ∠(zq, qp) ≥ π/2. Let x be the midpoint of thegeodesic 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 sep-arates p from A.

Proposition 7.10. A Cartan-Hadamard manifold N satisfies the separation prop-erly if and only if for every closed subset A and monotone decreasing sequence ofclosed subsets Ai in N such that ∩∞i=1Ai = A, then

∩∞i=1 C(Ai) = 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 inR2, then K∩S∞(R2) consists of the two poles in S1. However, C(K) = R2 becauseevery line given by x = constant is a geodesic joining the two poles of S1. Hence,C(K)∩S∞(R2) = S1. On the other hand, if we assume in addition that N satisfiesthe following separation property at infinity, then

C(K) ∩ S∞(N) = K ∩ S∞(N).

Definition 7.11. Let N be a Cartan-Hadamard manifold. N is said to satisfythe separation property at infinity if for any closed subset A of S∞(N) and anypoint p ∈ S∞(N) \ A, there exists a closed convex subset C in N such that A iscontained in the interior of C ∩ S∞(N) and p not in C.

216 Peter Li

It is easy to check that a two-dimensional visibility manifold alway satisfiesseparation property at infinity. On the other hand, upon improving a result ofM. Anderson [An], A. Borbely [Bo] has shown that Cartan-Hadamard manifold Nhas separation property at infinity provided that its sectional curvature satisfies−Ceαr(x) ≤ KN (x) ≤ −1 for some constant C > 0 and 0 ≤ α < 1/3, where r(x)is the distance from point x to a fixed point o ∈ N . We have the following simpleproposition.

Proposition 7.12. Let N be a Cartan-Hadamard manifold. Then for every closedset K in N ,

C(K) ∩ S∞(N) = K ∩ S∞(N)

if and only if N satisfies the separation property at infinity.

Theorem 7.13 (Li-Wang). Let M be a complete Riemannian manifold suchthat the dimension of the space of bounded harmonic functions H0(M) is h0(M).Let u : M → Hn be a harmonic map from M into a Cartan-Hadamard manifold,Nn. Denote A = u(M)∩S∞(N), where S∞(N) = Sn−1 is the geometric boundaryof N . Then there exists a set of points yiki=1 ⊂ u(M)∩N with k ≤ h0(M), suchthat,

u(M) ⊂ ∩iC(Ai ∪ yiki=1),

where Ai is a monotonically decreasing sequence of closed subsets of N properlycontaining A and ∩iAi = A. In addition, if we assume that N has the separationproperty, then

u(M) ⊂ C(A ∪ yiki=1).

As a corollary, we recovered the theorems of Cheng [Cg2] and Kendall [Kl]

Corollary 7.14 (Cheng, Kendall). If h0(M) = 1 and N is a Cartan-Hadamardmanifold, then every bounded harmonic map M → N must be constant.

Theorem 7.15 (Li-Wang). Suppose M is a complete manifold satisfying

h0(M) <∞

and all positive harmonic functions are bounded. Assume that u : M → N is aharmonic map from M into a Cartan-Hadamard manifold N which either is atwo-dimensional visibility manifold or has strongly negative sectional curvature,and that A = u(M) ∩ S∞(N) consists of at most one point. Then the set A isnecessarily empty, and there exists a set of k points yiki=1 ⊂ u(M) ∩ N withk ≤ h0(M) such that

u(M) ⊂ C(yiki=1).

In particular, if M has no nonconstant positive harmonic functions, then everysuch harmonic map must be a constant map.

Notice that the horoball of a visibility manifold intersects the geometricboundary at exactly one point (see [BGS]). Thus, we obtain the following Liouvilletype theorem which partially generalizes the results in [Sh] and [T].


Corollary 7.16. Suppose M does not admit any non-constant positive harmonicfunctions. Assume that N either is a two-dimensional visibility manifold or hasstrongly negative sectional curvature. Then every harmonic map from M into ahoroball of N must be constant.

Recall that a manifold is parabolic if it does not admit a positive Green’sfunction. It is well-known that a parabolic manifold has no massive subsets andevery positive harmonic function must be constant. Applying Theorem 7.13 tothis case, we have the following corollary.

Corollary 7.17. Let u be a harmonic map from a parabolic manifold M intoNn. Assume that N either is a two-dimensional visibility manifold or has stronglynegative sectional curvature. Then u(M) ⊂ C(A), where A = u(M) ∩ S∞(N).

Theorem 7.18 (Li-Wang). Let M be a complete manifold such that the maxi-mum number of disjoint d-massive sets of M is md(M). Suppose u : M → N is aharmonic map from M into N, and N satisfies the separation property at infinity.Assume that there exists a point o ∈ N such that r(u(x), o) = O(rd(x)) as x→∞.Then

A = u(M) ∩ S∞(N) = aik′

i=1

with k′ ≤ md(M)−m0(M). If, in addition, N either is a two-dimensional visibilitymanifold or has strongly negative sectional curvature, then there exist k pointsyjkj=1 ⊂ u(M) ∩N with k′ + k ≤ md(M) such that

u(M) ⊂ C(aik′i=1 ∪ yjkj=1).

Corollary 7.19. Let M be a complete manifold satisfying condition (M) and itsvolume growth Vp(R) = O(Rµ) for some point p ∈ M . Suppose N is a Cartan-Hadamard manifold satisfying either one of the following conditions:

(1) it has strongly negative sectional curvature;(2) it is a two-dimensional visibility manifold.

Let u : M → N be a harmonic map and suppose that there exists a point o ∈ Nsuch that r(u(x), o) = O(rd(x)) as x → ∞. Then there exist sets of k′ pointsaik

′

i=1 = u(M)∩S∞(N) and k points yjkj=1 ⊂ u(M)∩N with k′+k ≤ λ3(2d+µ)

such thatu(M) ⊂ C(aik

′i=1 ∪ yjkj=1).

If M is further assumed to have property (V), then we have k′ + k ≤ Cdµ−1.

We would like to remark that although Proposition 7.10 states that separationproperty is necessary and sufficient to conclude

∩∞i=1(Ai) = (A),

the proof of Theorem 7.13, only used the fact that ∩∞i=1(Ai) is bounded distancefrom (A). With this in mind, using a theorem of Anderson and Borbely, Theorem

218 Peter Li

7.13 and hence all consequential theorems of this section is valid for N being astrongly negatively curved manifold. A complete treatment can be found in [LW1].We would also like to recommend the interested reader to the paper by Han, Tam,Treibergs, and Wan [HTTW] where they consider harmonic maps from R2 intoH2 and obtained structural theorems of similar flavor but in a more detail fashionin terms of the Hopf differentials.

Finally, we like to also point out that the notion of massive sets also playeda role in understanding disjoint minimal graphs in RN . We will refer the reader to[LW7] for more details.

8 Lq Harmonic FunctionsIn this section, we will first present a basic estimate of Yau [Y2] for nonneg-

ative subharmonic functions. Since the absolute value of a harmonic function is asubharmonic function in the weak sense, this estimate is important in the studyof harmonic functions.

Lemma 8.1 (Yau). Let M be a complete manifold and f a nonnegative subhar-monic function defined on M. For any α > 0, and for any 0 ≤ R1 < R2 < R3 <

R4 <∞ we have the estimates∫Bp(R3)\Bp(R2)

fα−1 |∇f |2

≤ 4α2

((R2 −R1)−2

∫Bp(R2)\Bp(R1)

fα+1 + (R4 −R3)−2

∫Bp(R4)\Bp(R3)

fα+1

)

and ∫Bp(R3)

fα−1 |∇f |2 ≤ 4α2

(R4 −R3)−2

∫Bp(R4)\Bp(R3)

fα+1.

In [Y2], Yau used the above lemma to conclude that a complete manifolddoes not admit Lq harmonic functions for q > 1.

Theorem 8.2 (Yau). Let M be a complete manifold and f is a nonnegativesubharmonic function defined on M. Suppose for some 1 < q <∞ the Lq-norm off over balls of radius R satisfy the bound∫

Bp(R)

fq ≤ o(R2)

as R → ∞, then f must be a constant function. In particular, if M has infinitevolume and f ∈ Lq(M), then f must be identically 0.

Applying the theorem to the absolute value of a harmonic function, the fol-lowing corollary is a direct consequence of Theorem 8.2.


Corollary 8.3. A complete manifold does not admit any nonconstant Lq har-monic functions for 1 < q <∞.

Note that when q = ∞ the corollary is not valid because the Poincare diskhas infinitely many nonconstant bounded harmonic functions. It turns out thatwhen q ≤ 1 the corollary is also not valid without any extra assumption on themanifold. In this section, we will discuss the validity of this Liouville property forLq harmonic functions for the values of q ∈ (0, 1]. The condition to ensure Liouvileproperty for the case when q ∈ (0, 1) is quite different from the condition for thecase q = 1. Both of these cases was first considered by Li and Schoen [LS], wherethey obtained the sharp curvature condition for q ∈ (0, 1). In the same paper, theyalso obtained a condition for q = 1, while the sharp version was later proved by Liin [L3] using the heat equation method. Theorem 8.4 presented below is from [LS],while Theorem 8.6 is from [L3]. Counter-examples for both theorems when thecurvature conditions has been violated were given in [LS] indicating the sharpnessof these conditions.

Theorem 8.4 (Li-Schoen). Let Mm be a complete manifold. There exists aconstant δ(m) > depending only on m, such that, if the Ricci curvature of Msatisfies the lower bound

RicM (x) ≥ −δ(m) (1 + r(x))−2,

where r(x) is the distant to a fixed point p ∈ M , then any nonnegative, Lq sub-harmonic function f must be identically zero for 0 < q < 1.

Corollary 8.6. Let Mm be a complete manifold. There exists a constant δ(m) >depending only on m, such that if the Ricci curvature of M satisfies the lower

RicM (x) ≥ −δ(m) r−2(x)

then M does not admit any trivial Lq harmonic functions for 0 < q < 1.

Theorem 8.7 (Li). Let Mm be a complete manifold. Suppose the Ricci curvatureof M satisfies the lower

RicM (x) ≥ −C (1 + r(x))2

for some constant C > 0. Then any nonnegative, L1 subharmonic function mustbe constant.

Corollary 8.8. Let Mm be a complete manifold. Suppose the Ricci curvature ofM is bounded from below by

RicM (x) ≥ −C (1 + r(x))2

for some constant C > 0. Then M does not admit any nonconstant L1 harmonicfunctions.

220 Peter Li

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HarmonicFunctionson CompleteRiemannianManifolds...functions on complete Riemannian manifolds. Since the seminal work of Yau [Y1] in 1985, where he proved a Liouville theorem for positive