Weighted Poincaré inequality and rigidity of complete manifolds

WEIGHTED POINCARÉ INEQUALITY AND RIGIDITY OF COMPLETE MANIFOLDS

B

P

LI

J

WANG

ABSTRACT. – We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green’s function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function.

©2007 Elsevier Masson SAS

RÉSUMÉ. – Nous prouvons des théorèmes de structure pour des variétés complètes telles que la courbure de Ricci soit minorée, et satisfaisant l’inégalité de Poincaré à poids. Nous obtenons une estimation optimale de la décroissance de la fonction de Green positive et minimale. Cette estimation, qui dépend seulement du poids de la fonction dans l’inégalité de Poincaré, produit un critère de parabolicité de composantes connexes à l’infini utilisant le poids de la fonction.

©2007 Elsevier Masson SAS

Contents

0 Introduction . . . 921

1 Weighted Poincaré inequality . . . 925

2 Decay estimate . . . 930

3 Geometric conditions for parabolicity and nonparabolicity . . . 941

4 Improved Bochner formula and metric rigidity . . . 944

5 Rigidity and nonparabolic ends . . . 950

6 Warped product metrics . . . 960

7 Parabolic ends . . . 968

8 Nonexistence results for parabolic ends . . . 976

Acknowledgements . . . 981

References . . . 981

0. Introduction

Understanding the relations among the curvature, the topology and the function theory is a central theme in Riemannian geometry. Typically, one assumes the curvature to be bounded

1The first author was partially supported by NSF Grant DMS-0503735.

2The second author was partially supported by NSF Grant DMS-0404817.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

by a constant so that the comparison theorems become available. The main focus of the current paper, however, is to go beyond this realm to consider manifolds with Ricci curvature bounded below by a function. We will establish some structure theorems for such manifolds satisfying the following Poincaré type inequality.

DEFINITION 0.1. – Let Mⁿ be an n-dimensional complete Riemannian manifold. We say thatM satisfies aweighted Poincaré inequalitywith a nonnegative weight functionρ(x), if the inequality

M

ρ(x)φ²(x)dV

M

|∇φ|²dV

is valid for all compactly supported smooth functionφ∈C_c^∞(M).

DEFINITION 0.2. – Let Mⁿ be an n-dimensional complete Riemannian manifold. We say thatMhas property (Pρ) if a weighted Poincaré inequality is valid onM with some nonnegative weight functionρ. Moreover, theρ-metric, defined by

ds²_ρ=ρ ds²_M is complete.

Letλ1(M)denote the greatest lower bound of the spectrum of the Laplacian acting on L² functions. Then the variational principle forλ1(M)asserts the validity of the Poincaré inequal- ity, i.e.,

λ₁(M)

M

φ²

M

|∇φ|²

for all compactly supported functionsφ∈C_c^∞(M).Obviously,Mhas property (Pρ) with weight function ρ=λ1(M) if λ1(M)>0. Hence the notion of property (Pρ) can be viewed as a generalization of the assumptionλ1(M)>0.

We would like to point out that the idea of consideringds²_ρ was first used by Agmon [1]

in his study of eigenfunctions for the Schrödinger operators. Indeed, we will employ some of the arguments from [1] in this paper. We also remark that the weighted Poincaré inequalities in various forms have appeared in many important issues of analysis and mathematical physics. In the interesting papers [5,6], Fefferman and Phong have considered the more general weighted Sobolev type inequalities for pseudodifferential operators.

In Section 1, we will demonstrate that a complete manifold is nonparabolic if and only if it satisfies a weighted Poincaré inequality with some weight functionρ.Moreover, many nonpar- abolic manifolds satisfy property (Pρ) and we will provide some systematic ways to find a weight function. However, we would like to point out that the weight function is obviously not unique.

It turned out that some of the crucial estimates we developed in [11] can be generalized to give a sharper version of Agmon’s estimate for the Schrödinger operators developed in [1]. We believe that these estimates are interesting in their own rights and the sharp form will find more geometric applications in our further investigation. The proof of this decay estimate will be given in Section 2.

In Section 3, we will give geometric conditions (involvingρ) for an end being nonparabolic or parabolic using the decay estimate obtained in Section 2. The conditions are parallel to what we have established for the case whenρ=λ1(M)in [11]. In Section 4, we recall a generalized Bochner formula for the gradient of a harmonic function. The equality case for this inequality

will be discussed. Note that the generalized Bochner formula was first used by Yau in [21] and the equality case was also used previously in [19,11].

In Section 5, we give the proof of a structure theorem (Theorem 5.2) for manifolds with property (Pρ). The Ricci curvature is assumed to satisfy the inequality

RicM(x)−n−1 n−2ρ(x)

for allx∈M. In this theorem, a growth assumption is needed for the weight functionρin terms of theρ-distance. This allows to account for those functions ρthat have different growth rate with respect to the background distance in different directions. In any case, ifρis bounded then the growth assumption is fulfilled. When the dimension ofM is at least 4, the assumption onρis rather mild. In particular, ifρis a nondecreasing function of the distance to a compact set (with respect to the background metric) then the growth assumption is automatically satisfied.

In Section 6, we will study when a warped-product situation will actually occur for specific choices ofρ. Whenρis an increasing function, we give examples of the warped-product scenario.

On the other hand, we will also prove in Theorem 6.3 that when lim inf_r→∞ρ(r) = 0, the warped-product scenario does not exist. Let us summarize the results in Sections 5 and 6 in the following theorem. We denote byS(R)the maximum value of√

ρover the geodesic ball of radiusRwith respect to theρ-metric centered at a fixed pointp.

THEOREM A. –Let Mⁿ be a complete manifold with dimension n4. Assume that M satisfies property (Pρ) for some nonzero weight functionρ0. Suppose the Ricci curvature ofM satisfies the lower bound

Ric_M(x)−n−1 n−2ρ(x) for allx∈M. Ifρsatisfies the growth estimate

lim inf

R→∞ S(R) exp

−n−3 n−2R

= 0, then either

(1) M has only one nonparabolic end;or

(2) Mhas two nonparabolic ends and is given byM=R×Nwith the warped product metric ds²_M =dt²+η²(t)ds²_N,

for some positive functionη(t), and some compact manifoldN. Moreover,ρ(t)is a function oftalone satisfying

(n−2)ηη⁻¹=ρ and

lim inf

x→∞ ρ(x)>0.

We will state the casen= 3separately since the curvature condition for counting parabolic and nonparabolic ends are the same in this case. Again, the condition on the growth rate ofρis given with respect to theρ-distance. When restricted to those weight functions that are nondecreasing functions of the distance to the compact set the growth assumption is simply subexponential

growth with respect to the background distance function. Combining with the results in Section 7 for parabolic ends, the3-dimension case can be stated as follows:

THEOREM B. –Let M³ be a complete manifold of dimension3. Assume that M satisfies property(Pρ)for some nonzero weight functionρ0. Suppose the Ricci curvature ofMsatisfies the lower bound

RicM(x)−2ρ(x) for allx∈M. Ifρsatisfies the growth estimate

lim inf

R→∞ S(R)R⁻¹= 0, then either

(1) M has only one end;

(2) Mhas two nonparabolic ends and is given byM =R×Nwith the warped product metric ds²_M=dt²+η²(t)ds²_N,

for some positive function η(t), and some compact manifold N. Moreover, ρ(t) is a function oftalone satisfying

ηη⁻¹=ρ and

lim inf

x→∞ ρ(x)>0; or

(3) M has one parabolic end and one nonparabolic end and is given byM=R×Nwith the warped product metric

ds²_M=dt²+η²(t)ds²_N,

for some positive function η(t), and some compact manifold N. Moreover, ρ(t) is a function oftalone satisfying

ηη⁻¹=ρ and

lim inf

x→∞ ρ(x)>0 on the nonparabolic end.

Also, for dimensionn4we proved the following:

THEOREM C. –Let Mⁿ be a complete manifold of dimension n4 with property (Pρ).

Suppose the Ricci curvature ofM satisfies the lower bound Ric_M(x)− 4

n−1ρ(x) for allx∈M. Ifρsatisfies the property that

x→∞lim ρ(x) = 0, thenM has only one end.

Finally, in the last section, Section 8, we prove a nonexistence result indicating that for a large class of weight functionsρ, namely whenρ is a function of the distance and satisfying (ρ⁻¹⁴)(r)0 for r sufficiently large, there does not exist a manifold with property (Pρ) satisfying

Ric_M(x)− 4 n−1ρ(x).

We also proved a theorem restricting the behavior of the warped product.

As we have pointed out earlier, ifλ1(M)>0,then one may takeρ=λ1(M). This special case of Theorems A and B have been the subject of our earlier work [11,12]. The results generalized the work of Witten–Yau [20], Cai–Galloway [2], and Wang [19], on conformally compact manifolds. Also, a result in the similar spirit of Theorem C is available for this special case (see [12]). However, it remains open to deal with more general functionsρin Theorem C.

We would also like to point out that a similar theory was proposed in [13] for Kähler manifolds where the assumption is on the holomorphic bisectional curvature instead.

1. Weighted Poincaré inequality

In this section, we will show that it is not difficult to find a weight function for most manifolds.

The following proposition gives a convenient way to construct a weight function. The argument uses a modified version of the Barta inequality for the first eigenvalue. Using this method, we will show that a manifold is nonparabolic if and only if there exists a nontrivial weight function. We will also give examples of using other methods to obtain a weight function for various manifolds.

PROPOSITION 1.1. –LetMbe a complete Riemannian manifold. If there exists a nonnegative functionhdefined onM, that is not identically0, satisfying

Δh(x)−ρ(x)h(x), then the weighted Poincaré inequality

M

ρ(x)φ²(x)

M

|∇φ|²(x)

must be valid for all compactly supported smooth functionφ∈C_c^∞(M).

Proof. –LetD⊂M be a smooth compact subdomain ofM. Let us denoteλ₁(ρ, D)to be the first Dirichlet eigenvalue onDfor the operator

Δ +ρ(x).

Letube the first eigenfunction satisfying

Δu(x) +ρ(x)u(x) =−λ₁(ρ, D)u(x) onD and

u(x) = 0 on∂D.

We may assume thatu0onD,and the regularity ofuasserts thatu >0in the interior ofD.

Integration by parts yields

D

uΔh−

D

hΔu=

∂D

u∂h

∂ν −

∂D

h∂u

∂ν 0, (1.1)

whereν is the outward unit normal of∂D. On the other hand, the assumption onhimplies that uΔh−hΔuλ₁(ρ, D)uh.

Since bothu >0andhare not identically 0, this combining with (1.1) implies thatλ1(ρ, D)0.

In particular, the variational characterization ofλ1(ρ, D)implies that 0λ1(ρ, D)

D

φ²(x)

D

|∇φ|²−

D

ρ(x)φ²(x)

for all φ with support in D. Since D is arbitrary, this implies the weighted Poincaré inequality. 2

Let us now quickly recall the definition of parabolicity. Full detail on the discussion of parabolicity can be found in [8].

DEFINITION 1.2. – A complete manifold M is said to be nonparabolic if there exists a symmetric positive Green’s function G(x, y) for the Laplacian acting on L² functions.

Otherwise, we say thatM is parabolic.

Similarly, the notion of parabolicity is also valid when localized at an end of a manifold. We recall (see [8]) that an end is simply an unbounded component ofM\Ω, whereΩis a compact smooth domain ofM.

DEFINITION 1.3. – LetEbe an end of a complete manifoldM. We say thatEis nonparabolic if there exists a symmetric, positive, Green’s functionG(x, y)for the Laplacian acting on L² functions with Neumann boundary condition on∂E. Otherwise, we say thatE is a parabolic end.

Note that (see [8]) a complete manifold is nonparabolic if and only ifM has at least one nonparabolic end. More importantly, it is possible for a nonparabolic manifold to have many parabolic ends. It is also known that [8] a manifold is parabolic if we consider the sequence of harmonic functions{fi}defined onB(Ri)\B(R0)satisfying

Δf_i= 0 onB(R_i)\B(R₀) (1.2)

with boundary conditions

f_i=

1 on∂B(R0), 0 on∂B(Ri), (1.3)

then they converge to the constant functionf = 1defined onM \B(R0)as Ri→ ∞for any fixedR0.

COROLLARY 1.4. –LetMbe a complete nonparabolic manifold andG(p, x)be the minimal positive Green’s function defined onM with a pole at the pointp∈M. ThenM satisfies the weighted Poincaré inequality with the weight functionρgiven by

ρ(x) =|∇G(p, x)|² 4G²(p, x) .

Conversely, if a nonzero weight functionρ0exists, thenM must be nonparabolic.

Proof. –Let us first assume that M is nonparabolic and hence a positive symmetric Green’s function,G(x, y),for the Laplacian exists. Letp∈M be a fixed point and

g_a(x) = min

a, G(p, x) .

Thengais a superharmonic function defined onM. A direct computation yields that Δga¹²−|∇g_a|²

4g_a² ga¹²

onM. Hence Proposition 1.1 asserts that the weighted Poincaré inequality is valid with ρ=|∇g_a|²

4g²_a . Lettinga→ ∞,we conclude that we can take

ρ(x) =|∇G(p, x)|² 4G²(p, x) . This proves the first part of the corollary.

Conversely, let us assume that the weighted Poincaré inequality is valid for a nonzero weight function ρ0. Assuming on the contrary that M is parabolic, we will find a contradiction.

Indeed, ifM is parabolic then let us consider the sequence of compactly supported functions

φi=

⎧⎨

⎩

1 onB(R0),

fi onB(Ri)\B(R0), 0 onM\B(Ri)

wherefiis given by the sequence of harmonic functions obtained from (1.2) and (1.3). Setting φ=φiin the weighted Poincaré inequality, we have

M

ρφ²_i

M

|∇φi|²

=

B(Ri)\B(R0)

|∇f_i|²

=

∂B(Ri)

fi

∂fi

∂ν −

∂B(R0)

fi

∂fi

∂ν

=−

∂B(R0)

∂f_i

∂ν. However, sincefi→1onM\B(R0), we conclude that

M

ρ0,

violating the assumption thatρ0is nonzero. This proves the second part of the corollary. 2 While the existence of a weighted Poincaré inequality is equivalent to nonparabolicity, the condition thatM has property(Pρ)is not as clear cut. The following lemma gives a sufficient condition for(Pρ).

LEMMA 1.5. –LetMbe a nonparabolic complete manifold. SupposeG(x, y)is the minimal, symmetric, positive, Green’s function for the Laplacian acting onL²functions. For a fixed point p∈M, ifG(p, x)→0asx→ ∞, thenM has property(Pρ)with

ρ=|∇G(p, x)|² 4G²(p, x) .

Proof. –In view of Corollary 1.4, it remains to show that theρ-metric is complete. Indeed, letγ(s)be a curve parametrized by arclength,0sT. The length ofγ with respect to the ρ-metric is given by

γ

√ρ ds=1 2

γ

|∇logG|ds.

However, since logG

p, γ(0)

−logG

p, γ(T)

=− T 0

∂

∂s logG

p, γ(s) ds

γ

|∇logG|ds,

we conclude that theρ-length ofγis infinity ifγ(T)→ ∞. This proves the completeness of the ρ-metric. 2

We should point out that it is not necessarily true that ifG(p, x)does not tend to0at infinity thenM does not have property (Pρ) since there might be another weight function that gives a complete metric.

Example 1.6. – WhenM=Rⁿforn3,the Green’s function is given by G(0, x) =Cnr²⁻ⁿ(x)

for some constantCn>0depending only onn. In this case, we compute that

|∇G(0, x)|²

4G²(0, x) =(n−2)² 4 r⁻²(x).

Hence from the above discussion, we conclude that the weighted Poincaré inequality (n−2)²

4

Rⁿ

r⁻²φ²

Rⁿ

|∇φ|²,

which is the well-known Hardy’s inequality, must be valid for all compactly supported smooth functionφ∈C_c^∞(Rⁿ), andRⁿhas property (Pρ) with

ρ(x) =(n−2)² 4 r⁻²(x).

Example 1.7. – LetMⁿbe a minimal submanifold of dimensionn3inR^N. If we denoter¯ to be the extrinsic distance function ofR^N to a fixed pointp∈M,then it is known that it satisfies the equation

Δ¯r(n−1)¯r⁻¹, (1.4)

where Δ is the Laplacian on M with respect to the induced metric from R^N. For any φ∈ C_c^∞(M), we consider the integral

M

¯

r⁻¹φ²Δ¯r=−2

M

¯

r⁻¹φ∇φ,∇¯r +

M

¯

r⁻²φ²|∇¯r|²

2

M

¯

r⁻¹φ|∇φ|+

M

¯ r⁻²φ². Combining with (1.4), this implies that

(n−2) 2

M

¯ r⁻²φ²

M

¯

r⁻¹φ|∇φ|

M

¯ r⁻²φ²

¹₂

M

|∇φ|^{2 1}₂

, hence the weighted Poincaré inequality is valid onM with

ρ=(n−2)² 4 r¯⁻².

If we further assume that M is properly immersed, then Lemma 1.5 implies that M has property(Pρ).

Example 1.8. – Let M be a simply connected, complete, Cartan–Hadamard manifold with sectional curvature bounded from above by

KM−1.

In this case, the Hessian comparison theorem asserts that Δr(n−1) cothr, (1.5)

whereris the geodesic distance function from a fixed pointp∈M. Supposeφ∈C_c^∞(M)is a compactly supported function. Then (1.5) and integration by parts yield

(n−1)

M

φ²cothr

M

φ²Δr

=−2

M

φ∇φ,∇r

2

M

φ|∇φ|

n−1 2

M

φ²+ 2 n−1

M

|∇φ|².

This can be rewritten as (n−1)²

4

M

φ²+(n−1)² 2

M

(cothr−1)φ²

M

|∇φ|².

Hence in this case, the weight functionρcan be taken to be ρ=(n−1)²

4 +(n−1)²

2 (cothr−1) (n−1)²

4 .

Since it is bounded from below by a positive constant, the ρ-metric must be complete by the completeness assumption onds²_M,andM has property(Pρ).

2. Decay estimate

In this section, we will consider a more general situation. LetV be a given potential function defined onM,and

Δ−V(x)

be the Schrödinger operator onM. We assume that there exists a positive functionρdefined on M, such that the weighted Poincaré type inequality

M

ρ(x)φ²(x)dx

M

|∇φ|²(x)dx+

M

V(x)φ²(x)dx (2.1)

is valid for any compactly supported functionφ∈C_c^∞(M). Let us define theρ-metric given by ds²_ρ=ρ ds².

Using this metric, we consider theρ-distance function defined to be, rρ(x, y) = inf

γ ρ(γ),

the infimum of the lengths of all smooth curves joiningxandywith respect tods²_ρ. For a fixed point p∈M, we denote rρ(x) =rρ(p, x) to be the ρ-distance to p. One checks readily that

|∇rρ|²(x) =ρ(x). As in the case whenV = 0,we say that the manifold has property(Pρ,V)if theρ-metric is complete, and this will be the standing assumption on thatM.

Throughout this article, we denote Bρ(p, R) =

x∈M|rρ(p, x)< R

to be the set of points inM that hasρ-distance less thanRfrom pointp. We also denote B(p, R) =

x∈M|r(p, x)< R

to be the set of points in M that has distance less than R from point p with respect to the background metric ds²_M. When p∈M is a fixed point, we will suppress the dependency of p and write B_ρ(R) =B_ρ(p, R) and B(R) =B(p, R). If E is an end of M, we denote E_ρ(R) =B_ρ(R)∩E.

THEOREM 2.1. –LetM be a complete Riemannian manifold with property(Pρ,V). Suppose E is an end of M such that there exists a nonnegative function ρ(x)defined on E with the property that

E

ρ(x)φ²(x)dx

E

|∇φ|²(x)dx+

E

V(x)φ²(x)dx

for any compactly supported functionφ∈C_c^∞(E). Letf be a nonnegative function defined on Esatisfying the differential inequality

Δ−V(x)

f(x)0.

Iffsatisfies the growth condition

Eρ(R)

ρf²exp(−2rρ) =o(R)

asR→ ∞, then it must satisfy the decay estimate

Eρ(R+1)\Eρ(R)

ρf²Cexp(−2R)

for some constantC >0depending onf andρ.

Proof. –We will first prove that for any 0< δ <1,there exists a constant0< C <∞such that,

E

ρexp(2δr_ρ)f²C.

Indeed, letφ(rρ(x))be a nonnegative cut-off function with support in Ewithrρ(x)being the ρ-distance to the fixed pointp. Then for any functionh(rρ(x))integration by parts yields

E

∇

φfexp(h)² (2.2)

=

E

∇

φexp(h)²f²+

E

φexp(h)2

|∇f|² + 2

E

φfexp(h)

∇

φexp(h) ,∇f

=

E

∇

φexp(h)²f²+

E

φ²exp(2h)|∇f|²

+1 2

E

∇

φ²exp(2h) ,∇

f²

=

E

∇

φexp(h)²f²+

E

φ²exp(2h)|∇f|²−1 2

E

φ²exp(2h)Δ f²

=

E

∇

φexp(h)²f²−

E

φ²exp(2h)fΔf

E

∇

φexp(h)²f²−

E

V φ²exp(2h)f²

=

E

|∇φ|²f²exp(2h) + 2

E

φexp(2h)∇φ,∇h f²

+

E

φ²|∇h|²f²exp(2h)−

E

V φ²f²exp(2h).

On the other hand, using the assumption (2.1), we have

E

ρφ²f²exp(2h)

E

∇

φfexp(h)²+

E

V φ²f²exp(2h),

hence (2.2) becomes

E

ρφ²f²exp(2h)

E

|∇φ|²f²exp(2h) + 2

E

φexp(2h)∇φ,∇h f² (2.3)

+

E

φ²|∇h|²f²exp(2h).

Let us now choose

φ rρ(x)

=

⎧⎪

⎪⎨

⎪⎪

⎩

rρ(x)−R0 onEρ(R0+ 1)\Eρ(R0), 1 onEρ(R)\Eρ(R0+ 1), R⁻¹(2R−rρ(x)) onEρ(2R)\Eρ(R),

0 onE\Eρ(2R),

and hence

|∇φ|²(x) =

⎧⎨

⎩

ρ(x) onEρ(R0+ 1)\Eρ(R0), R⁻²ρ(x) onEρ(2R)\Eρ(R),

0 on(Eρ(R)\Eρ(R0+ 1))∪(E\Eρ(2R)).

We also choose

h rρ(x)

=

δr_ρ(x) forr_ρ(1+δ)^K , K−r_ρ(x) forr_ρ(1+δ)^K , for some fixedK >(R0+ 1)(1 +δ). WhenR_(1+δ)^K , we see that

|∇h|²(x) =

δ²ρ(x) forrρ_(1+δ)^K , ρ(x) forr_ρ(1+δ)^K and

∇φ,∇h (x) =

⎧⎨

⎩

δρ(x) onE_ρ(R₀+ 1)\E_ρ(R₀), R⁻¹ρ(x) onE_ρ(2R)\E_ρ(R),

0 otherwise.

Substituting into (2.3), we obtain

E

ρφ²f²exp(2h)

Eρ(R0+1)\Eρ(R0)

ρf²exp(2h) +R⁻²

Eρ(2R)\Eρ(R)

ρf²exp(2h)

+ 2δ

Eρ(R0+1)\Eρ(R0)

ρf²exp(2h)

+ 2R⁻¹

Eρ(2R)\Eρ(R)

ρf²exp(2h)

+δ²

Eρ(K(1+δ)⁻¹)\Eρ(R0)

ρφ²f²exp(2h)

+

Eρ(2R)\Eρ(K(1+δ)⁻¹)

ρφ²f²exp(2h).

This can be rewritten as

Eρ(K(1+δ)⁻¹)\Eρ(R0+1)

ρf²exp(2h)

Eρ(K(1+δ)⁻¹)

ρφ²f²exp(2h)

Eρ(R0+1)\Eρ(R0)

ρf²exp(2h)

+R⁻²

Eρ(2R)\Eρ(R)

ρf²exp(2h)

+ 2δ

Eρ(R0+1)\Eρ(R0)

ρf²exp(2h)

+ 2R⁻¹

Eρ(2R)\Eρ(R)

ρf²exp(2h)

+δ²

Eρ(K(1+δ)⁻¹)\Eρ(R0)

ρf²exp(2h),

hence

1−δ²

Eρ(K(1+δ)⁻¹)\Eρ(R0+1)

ρf²exp(2h)

δ²+ 2δ+ 1

Eρ(R0+1)\Eρ(R0)

ρf²exp(2h)

+R⁻²

Eρ(2R)\Eρ(R)

ρf²exp(2h) + 2R⁻¹

Eρ(2R)\Eρ(R)

ρf²exp(2h).

The definition ofhand the assumption on the growth condition onfimply that the last two terms on the right-hand side tend to0asR→ ∞. Hence we obtain the estimate

1−δ²

Eρ(K(1+δ)⁻¹)\Eρ(R0+1)

ρf²exp(2δr_ρ)

δ²+ 2δ+ 1

Eρ(R0+1)\Eρ(R0)

ρf²exp(2δrρ).

Since the right-hand side is independent ofK, by lettingK→ ∞we conclude that

E\Eρ(R0+1)

ρf²exp(2δr_ρ)C, (2.4)

for some constant0< C <∞.

Our next step is to improve this estimate by settingh=rρin the preceding argument. Note that with this choice ofh, (2.3) asserts that

−2

E

φexp(2rρ)∇φ,∇rρ f²

E

|∇φ|²f²exp(2rρ).

ForR0< R1< R, let us chooseφto be

φ(x) =

_rρ(x)−R0

R1−R0 onE_ρ(R₁)\E_ρ(R₀),

R−rρ(x)

R−R1 onEρ(R)\Eρ(R1).

We conclude that 2 R−R1

Eρ(R)\Eρ(R1)

R−r_ρ(x) R−R1

ρf²exp(2rρ)

1

(R₁−R₀)²

Eρ(R1)\Eρ(R0)

ρf²exp(2rρ)

+ 1

(R−R1)²

Eρ(R)\Eρ(R1)

ρf²exp(2r_ρ)

+ 2

(R₁−R₀)²

Eρ(R1)\Eρ(R0)

(rρ−R0)ρf²exp(2rρ).

On the other hand, for any0< t < R−R₁, since 2t

(R−R1)²

Eρ(R−t)\Eρ(R1)

ρf²exp(2rρ)

2

(R−R₁)²

Eρ(R)\Eρ(R1)

R−rρ(x)

ρf²exp(2rρ),

we deduce that 2t (R−R1)²

Eρ(R−t)\Eρ(R1)

ρf²exp(2rρ) (2.5)

2

R1−R0

+ 1

(R1−R0)²

Eρ(R1)\Eρ(R0)

ρf²exp(2r_ρ)

+ 1

(R−R1)²

Eρ(R)\Eρ(R1)

ρf²exp(2rρ).

Observe that by takingR1=R0+ 1,t= 1, and setting g(R) =

Eρ(R)\Eρ(R0+1)

ρf²exp(2rρ),

the inequality (2.5) can be written as

g(R−1)C₁R²+1 2g(R), where

C₁=3 2

Eρ(R0+1)\Eρ(R0)

ρf²exp(2r_ρ)

is independent ofR. Iterating this inequality, we obtain that for any positive integerkandR1

g(R)C₁ k i=1

(R+i)²

2ⁱ⁻¹ + 2^−kg(R+k) C1R²

∞ i=1

(1 +i)²

2ⁱ⁻¹ + 2^−kg(R+k) C2R²+ 2^−kg(R+k)

for some constantC2. However, our previous estimate (2.4) asserts that

E

ρf²exp(2δrρ)C

for anyδ <1. This implies that g(R+k) =

Eρ(R+k)\Eρ(R0+1)

ρf²exp(2rρ)

exp

2(R+k)(1−δ)

Eρ(R+k)\Eρ(R0+1)

ρf²exp(2δrρ)

Cexp

2(R+k)(1−δ) . Hence,

2^−kg(R+k)→0 ask→ ∞by choosing2(1−δ)<ln 2. This proves the estimate

g(R)C2R². By adjusting the constant, we have

Eρ(R)

ρf²exp(2r_ρ)C₃R² (2.6)

for allRR₀.

Using inequality (2.5) again and by choosingR1=R0+ 1andt=^R₂ this time, we conclude that

R

Eρ(^R₂)\Eρ(R0+1)

ρf²exp(2rρ)C4R²+

Eρ(R)\Eρ(R0+1)

ρf²exp(2rρ).

However, applying the estimate (2.6) to the second term on the right-hand side, we have

Eρ(^R₂)\Eρ(R0+1)

ρf²exp(2r_ρ)C₅R.

Therefore, forRR₀,

Eρ(R)

ρf²exp(2r_ρ)CR.

(2.7)

We are now ready to prove the theorem by using (2.7). Settingt= 2andR1=R−4in (2.5), we obtain

Eρ(R−2)\Eρ(R−4)

ρf²exp(2rρ)

8

R−R0−4+ 4 (R−R0−4)²

Eρ(R−4)\Eρ(R0)

ρf²exp(2r_ρ)

+1 4

Eρ(R)\Eρ(R−4)

ρf²exp(2rρ).

According to (2.7), the first term of the right-hand side is bounded by a constant. Hence, the above inequality can be rewritten as

Eρ(R−2)\Eρ(R−4)

ρf²exp(2rρ)C+1 3

Eρ(R)\Eρ(R−2)

ρf²exp(2rρ).

Iterating this inequalityktimes, we arrive at

Eρ(R+2)\Eρ(R)

ρf²exp(2rρ)

C

k−1

i=0

3⁻ⁱ+ 3^−k

Eρ(R+2(k+1))\Eρ(R+2k)

ρf²exp(2r_ρ).

However, using (2.7) again, we conclude that the second term is bounded by 3^−k

Eρ(R+2(k+1))\Eρ(R+2k)

ρf²exp(2rρ)C3^−k

R+ 2(k+ 1)

which tends to0ask→ ∞. Hence

Eρ(R+2)\Eρ(R)

ρf²exp(2rρ)C (2.8)

for some constantC >0independent ofR. The theorem now follows from (2.8). 2 We now draw some corollaries.

COROLLARY 2.2. –LetM be a complete Riemannian manifold. SupposeEis an end ofM such thatλ₁(E)>0, i.e.,

λ₁(E)

E

φ²(x)dx

E

|∇φ|²(x)dx

for any compactly supported functionφ∈C_c^∞(E). Letf be a nonnegative function defined on Esatisfying the differential inequality

(Δ +μ)f(x)0

for some constantμwith the property thatλ1(E)−μ >0. Ifa=

λ1(E)−μandf satisfies the growth condition

E(R)

f²exp(−2ar) =o(R) asR→ ∞, then it must satisfy the decay estimate

E(R+1)\E(R)

f²Cexp(−2aR)

for some constantC >0depending onf anda.

Proof. –By setting−V(x) =μwe can rewrite the Poincaré inequality as λ1(E)−μ

E

φ²(x)dx

E

|∇φ|²(x)dx−μ

E

φ²(x)dx.

We now can apply Theorem 2.1 by settingρ=a². The distance function with respect to the metricρ ds²is then given by

r_ρ(x) =ar(x)

wherer(x)is the background distance function to the smooth compact setΩ⊂M. The corollary follows from Theorem 2.1. 2

COROLLARY 2.3. –Let M be a complete Riemannian manifold satisfying property (Pρ).

Suppose{E1, . . . , Ek} withk2are the nonparabolic ends of M. Then for each1ik there exists a bounded harmonic functionf_idefined onM satisfying the growth estimate

Bρ(R+1)\Bρ(R)

|∇f_i|²Cexp(−2R).

Moreover,0fi1and has the property that

x∈Esupi

fi(x) = 1, and

x∈Einfj

fi(x) = 0, forj=i.

Proof. –We will constructf_ifor the casei= 1, and the construction for other values ofiis exactly the same. In this case, we will simply denotef=f1. Following the theory of Li–Tam [10]

(see also [11]),fcan be constructed by taking the limit, asR¯→ ∞, of a converging subsequence of harmonic functionsfR¯satisfying

ΔfR¯= 0 onB( ¯R), with boundary condition

fR¯= 1 on∂B( ¯R)∩E1,

and

fR¯= 0 on∂B( ¯R)\E1.

In fact, we only need to verify the growth estimate for the Dirichlet integral for the limiting function. The other required properties off follow from the construction of Li–Tam. To check the growth estimate, we first show that on(Bρ(R+ 1)\Bρ(R))\E1, because of the boundary condition we can apply Theorem 2.1 to the functionfR¯. By taking the limit, this implies that

(Bρ(R+1)\Bρ(R))\E1

ρf²Cexp(−2R).

(2.9)

Similarly onE₁,we can apply Theorem 2.1 to the function1−fR¯,hence we obtain

(E1∩Bρ(R+1))\(E1∩Bρ(R))

ρ(1−f)²Cexp(−2R).

(2.10)

Let us now consider the cut-off function

φ rρ(x)

=

⎧⎪

⎨

⎪⎩

rρ(x)−R+ 1 forR−1rρR,

1 forRr_ρR+ 1,

R+ 2−r_ρ forR+ 1r_ρR+ 2,

0 otherwise.

Integrating by parts and Schwarz inequality yield 0 =

(Bρ(R+2)\Bρ(R−1))\E1

φ²fΔf

=−

(Bρ(R+2)\Bρ(R−1))\E1

φ²|∇f|²−2

(Bρ(R+2)\Bρ(R−1))\E1

φf∇φ,∇f

−1

2

(Bρ(R+2)\Bρ(R−1))\E1

φ²|∇f|²+ 2

(Bρ(R+2)\Bρ(R−1))\E1

|∇φ|²f².

Hence combining with the definition ofφ, we obtain the estimate

(Bρ(R+1)\Bρ(R))\E1

|∇f|²

(Bρ(R+2)\Bρ(R−1))\E1

φ²|∇f|²

4

(Bρ(R+2)\Bρ(R−1))\E1

ρf².

Applying the estimate (2.9) to the right-hand side, we conclude the desired estimate on the set (B_ρ(R+ 1)\B_ρ(R))\E₁. The estimate on(E₁∩B_ρ(R+ 1))\(E₁∩B_ρ(R))follows by using the function1−f and (2.10) instead. 2

We would like to point out that the hypothesis of Corollary 2.2, hence Theorem 2.1, is best possible. Indeed, if we consider the hyperbolic space formHⁿof−1constant sectional curvature, then the volume growth is given by

V(R)∼Cexp

(n−1)R

and

λ1

Hⁿ

=(n−1)²

4 .

We consider Theorem 2.1 for the special case whenV(x) = 0andρ=⁽ⁿ⁻₄¹⁾². In this case, the distance functionr_ρwith respect to the metricρ ds²is simply given by

r_ρ=(n−1) 2 r,

whereris the hyperbolic distance function. Iff is a nonconstant bounded harmonic function,

then

B(R)

ρf²exp(−2rρ) =O(R).

We claim that the conclusion of Theorem 2.1 is not valid, hence will imply that the hypothesis of Theorem 2.1 cannot be improved. Indeed, if the conclusion were true, thenf would be in L²(Hⁿ). However, Yau’s theorem [22] implies thatf must be identically constant. On the other hand, it is known that Hⁿ has an infinite dimensional space of bounded harmonic functions, which provides a contradiction.

Also note that in the case ofRⁿ (n3), the distance function with respect to theρ-metric is

rρ∼n−2 2

r 1

t⁻¹dt= logrⁿ⁻²²

asr→ ∞. If we consider a multiple of Green’s functionf(x) =r²⁻ⁿonRⁿ, then checking the hypothesis of Theorem 2.1 forf(x)onE=Rⁿ\B(1),the integral

E

ρf²exp(−2r_ρ) =(n−2)² 4

∞ 1

r⁻²r⁻ⁿ⁺²r⁴⁻²ⁿrⁿ⁻¹dr

=(n−2)² 4

∞ 1

r⁻²ⁿ⁺³dr

<∞.

Hence we can apply Theorem 2.1 to this choice off. On the other hand, the integral

Eρ(R+1)\Eρ(R)

ρf²=(n−2)² 4

rρ=R rρ=R+1

r⁻²r⁴⁻²ⁿrⁿ⁻¹dr

=

e^n−22R

e

2(R+1) n−2

r⁻ⁿ⁺¹dr

= 1

n−2

exp(−2R)−exp

−2(R+ 1)

=1−e⁻²

n−2 exp(−2R).

This implies that the conclusion of Theorem 2.1 is also sharp in this case.

Finally, we point out that the preceding argument of Theorem 2.1 can be extended without much modification to deal withp-forms satisfying a suitable differential equation. We consider the operator

Δ +W(x)

acting on thep-forms onM,whereΔis the Hodge Laplacian andW an endomorphism on the bundle ofp-forms onM.

THEOREM 2.4. –LetMbe a complete Riemannian manifold. SupposeEis an end ofMsuch that there exists a nonnegative functionρ(x)defined onEwith the property that

E

ρ(x)|η|²(x)dx

E

|dη|²(x) +|δη|²(x) dx+

E

W(η)(x), η(x) dx

is valid for any compactly supported smoothp-formηonE. Assume that theρ-metric given by ds²_ρ=ρ ds²_Mis complete onE. Letωbe a smoothp-form defined onEsatisfying the differential inequality

Δ +W(x) ω, ω

(x)0 for allx∈E. Ifωsatisfies the growth condition

Eρ(R)

ρ|ω|²exp(−2r_ρ) =o(R)

asR→ ∞, then it must satisfy the decay estimate

Eρ(R+1)\Eρ(R)

ρ|ω|²Cexp(−2R)

for some constantC >0depending onωandρ.

3. Geometric conditions for parabolicity and nonparabolicity

In this section, we would like to discuss some geometric conditions for the parabolicity and nonparabolicity of an endE. In [11], we used the decay estimate similar to Section 2 to derive geometric conditions for parabolicity on a manifold with λ1(M)>0. A similar argument will yield the following conditions for manifolds with property (Pρ). The key issue is that when (Pρ) is present, the geometric conditions involvingρfor parabolicity and nonparabolicity has a substantial gap. This fact is important to the proof of our main theorems in the proceeding sections.

THEOREM 3.1. –LetEbe an end of a complete Riemannian manifoldMwith property(Pρ) for some weight functionρ. IfEis nonparabolic, then

Eρ(R+1)\Eρ(R)

ρ dV C₁exp(2R)