Generalized Liouville property for Schrödinger operator on Riemannian manifolds

Digital Object Identifier (DOI) 10.1007/s002090100257Math. Z. 238, 355–387 (2001)

Generalized Liouville property for Schrodingeroperator on Riemannian manifolds

Seok Woo Kim1, Yong Hah Lee2

1 School of Mathematics, Korea Institute for Advanced Study, Seoul 130–012, Korea(e-mail: [email protected])

2 Department of Mathematics Education, EwhaWomans University, Seoul 120–750, Korea(e-mail: [email protected])

Received: 4 April 2000; in final form: 19 September 2000 /Published online: 25 June 2001 –c© Springer-Verlag 2001

Abstract. In this paper, we prove that the dimension of the space of positive(bounded, respectively)L-harmonic functions on a complete RiemannianmanifoldwithL-regular ends is equal to thenumberof ends (L-nonparabolicends, respectively). This result is asolutionof anopenproblemofGrigor’yanrelated to the Liouville property for the Schrodinger operatorL. We alsoprove that if a given complete Riemannian manifold is roughly isometric toa complete Riemannian manifold satisfying the volume doubling condition,the Poincare inequality and the finite covering condition on each end, thenthe dimension of the space of positive (bounded, respectively,) solutions forthe Schrodinger operator with a potential satisfying a certain decay rate onthe manifold is equal to the number of ends (L-nonparabolic ends, respec-tively). This is a partial answer of the question, suggested by Li, related tothe regularity of ends of a complete Riemannian manifold. Especially, ourresults directly generalize various earlier results of Yau, of Li and Tam, ofGrigor’yan, and of present authors, but with different techniques that thepeculiarity of the Schrodinger operator demands.

1 Introduction

The Liouville type theorem for an elliptic operator on a complete Rieman-nian manifold has long been an interesting topic of study to analysts andgeometers. Given a classF of functions and an elliptic operatorL on a com-plete Riemannianmanifold, by the Liouville property of the pair(F ,L), we

The first author is supported in part by GARC and BSRI.

356 S.W. Kim, Y.H. Lee

mean that the dimension of the space of all solutions for the elliptic operatorL in F is at most one. In this viewpoint, it is natural to regard the finite di-mensionality of the solution spaceF as the generalized Liouville propertyof the pair(F ,L). For instance, whenL is the Laplacian∆ andF is theclass of positive harmonic functions on a complete Riemannian manifold,it is well understood by the works of Yau [21], Grigor’yan [5], [8], Li andTam [17], [18], the present authors [14] and others.

In a series of papers [6] and [8], Grigor’yan gave interesting resultsrelated to the Liouville property for bounded solutions for the SchrodingeroperatorL = ∆− V with nontrivial potentialV . In [6], he proved that if acomplete RiemannianmanifoldM is parabolic for the Laplacian, then everybounded solution for the Schrodinger operatorL must be identically zero.On the other hand, Grigor’yan [5] proved the generalized Liouville propertyrelated to the regularity as follows:

Theorem 1 LetM be a complete Riemannian manifold with regular ends.Suppose thatM has nonparabolic endsE1, E2, · · · , El, l ≥ 1, and parabo-lic endse1, e2, · · · , es, s ≥ 0. Then for any nonnegative harmonic functionu onM , the nonnegative numbers

ai = limx→∞,x∈Ei

u(x) and bj = fluxej u (1)

exist for eachi = 1, 2, · · · , l andj = 1, 2, · · · , s. Conversely, given nonneg-ative numbersa1, a2, · · · , al, b1, b2, · · · , bs, there exists a unique harmonicfunctionu onM satisfying (1) for eachi = 1, 2, · · · , l andj = 1, 2, · · · , s.

Similarly, every bounded harmonic functionu onM is uniquely char-acterized by real numbersa1, a2, · · · , al such thatai = lim

x→∞,x∈Ei

u.

In [8], Grigor’yan suggested an open problem:Can one state an analogueof Theorem 1 for the Schrodinger operator? In this paper, motivated byGrigor’yan’s open problem [8] and his previous works [5], [6] and [8],we develop a theory of the Schrodinger operatorL on a class of completeRiemannian manifolds in the following setting: We say that a functionu ona complete RiemannianmanifoldM is anL-harmonic function ifu satisfiesthe following Schrodinger equation

Lu = (∆− V )u = 0,

where∆ is the Laplacian ofM andV is a nonnegative function onM .This equation is understood in the distribution sense. For convenience’sake, throughout this paper, we assume that every potentialV is contin-uous. From this assumption, one can achieve the continuity of solutions ofthe Schrodinger equation. More generally, such a result can be extended topotentials in the local Kato class. (See [9].) From now on, when we say that

Generalized Liouville property for Schrodinger operator 357

L = ∆ − V is a Schrodinger operator on a complete Riemannian mani-foldM ,∆ denotes the Laplacian ofM andV is a nonnegative continuousfunction onM .

We introduce some new concepts needed in deploying our theory: Allendsare classifiedby theL-nonparabolicity,whichmeans that thereexists anL-harmonic functionuE , called anL-harmonic measure, on an endE suchthat0 ≤ uE ≤ 1 onE,uE = 0 on∂E andsupE uE = 1.Wealso define theL-regularity of eachendE, in Sect. 3, which asserts the validity of the sphereversion of the Harnack inequality for positiveL-harmonic functions onE.In fact, theL-nonparabolicity and theL-regularity enable one to describeexactly the behavior near infinity of boundedL-harmonic functions. Thefollowing theorem shows a characterization of theL-parabolicity:Theorem 2 (Strong Liouville Theorem) LetM have onlyL-parabolic(not necessarilyL-regular) ends. Then every positiveL-harmonic functionand every boundedL-harmonic function onM must be zero.

We also give an affirmative answer to Grigor’yan’s open problem. Toour knowledge, this is the first nontrivial theorem, which concerns the finitedimensionality of the space of positive (bounded, respectively)L-harmonicfunctions:

Theorem 3 Let M be a complete Riemannian manifold withL-regularends and have at least oneL-nonparabolic end. Suppose thatM hasL-nonparabolic endsE1, E2, · · · , El and L-parabolic endse1, e2, · · · , es.Then every positiveL-harmonic function is uniquely represented byfE1 , fE2 , · · · , fEl

,he1 , he2 , · · · , hes , i.e., for eachpositiveL-harmonic func-tion f , there exist positive constantsa1, a2, · · · , al and b1, b2, · · · , bs suchthat

f =l∑

i=1

aifEi +s∑

j=1

bjhej ,

wherefE1 , fE2 , · · · , fEl, he1 , he2 , · · · , hes are positiveL-harmonic func-

tions, explained in Sect. 4. In particular, all boundedL-harmonic functionsare generated byfE1 , fE2 , · · · , fEl

, i.e., for each boundedL-harmonic func-tion f , there exist constantsa1, a2, · · · , al such that

f =l∑

i=1

aifEi .

Furthermore, the setfEi , hej : i = 1, 2, · · · , l, j = 1, 2, · · · , s is linearlyindependent. Therefore, we have

dimH+L (M) = s+ l, and dimHBL(M) = l,


whereH+L (M) and HBL(M) denote the space spanned by positiveL-

harmonic functions and the space of boundedL-harmonic functions onM ,respectively.

Theorem 3 is a rather delicate result. The main point is that one cannotuse the well known procedure for harmonic functions due to the potentialterm. For example, in the case of harmonic functions, every constant is alsoa solution. Therefore, we may regard every bounded harmonic function as apositive harmonic function taking the value zero as its infimum. However inthe case of the Schrodinger operator with nontrivial potential, each nonzeroconstant is not a solution any longer. So such techniques as adding or sub-tracting a constant from a solution to get another one need to be modified ina subtle way. Another difficulty one encounters is that the minimum princi-ple for the Schrodinger operator is not valid, i.e., a positive solution for theSchrodinger operator may admit an interior minimum.

The key argument in proving Theorem 3 is to construct variousL-harmonic functions, which roughly take over the role of barriers at infinity.In Sect. 2, we define anL-harmonic functionLE on eachL-nonparabolicendE, called the Liouville function of the endE. The Liouville functionLE of E, together with theL-harmonic measureuE , plays a crucial role inanalyzing behaviors near infinity ofL-harmonic functions of the endE. Wealso construct the Liouville functionL on the whole manifoldM such thatsupM L = 1, in particular,L has the same behavior near infinity of eachL-nonparabolic endE as that of theL-harmonic measureuE .

Through Sect. 3 and Sect. 4, we prove that each of positiveL-harmonicfunctions and boundedL-harmonic functions onM has the same behaviornear infinity of eachL-nonparabolic endE as that ofcLE , hence that ofcL,for some constantc ∈ R. It may be curious to state the behavior near infinityof ends ofL-harmonic functions in such a way. But this is in fact in linewith the approach of [5] and [8]. It is reasonable to state the asymptoticallyconstant behavior of harmonic functions, because each constant function isstill a solution for the Laplacian∆. However, in the case of the Schrodingeroperator∆− V with nontrivial potentialV , each nonzero constant is not asolution any longer. Therefore, in order to describe the behavior near infinityof solutions for the Schrodinger operator, it is needed to lean on the behaviorof the Liouville functionLE orL.

InSect. 5,wegiveanexampleof theL-regular end. In [15], Li proved thatif the volumedoubling condition, thePoincare inequality and theSobolev in-equality are satisfied on a complete Riemannianmanifold, then the Harnackinequality for nonnegativesolutions for theSchrodingeroperatorL = ∆−Vholdsoneachball, butwith theHarnackconstant dependingon the radiusandthe potentialV . On the other hand, Hajl´ asz and Koskela [10] obtained theSobolev inequality under the volume doubling condition and the Poincare


inequality. Thus from these results, we get the Harnack inequality for theSchrodinger operatorL = ∆−V as follows: Leto be a fixed point of a com-plete Riemannian manifoldM andE be an end ofM satisfying the volumedoubling condition(V D)∞ and the Poincare inequality(P )∞, explained inSect. 5. Then for any pointx ∈ ∂BR(o) ∩ E and any0 < r < R/2, thereexists a positive constantC depending on((supBr(x) V )r2 + 1) such thatfor any nonnegativeL-harmonic functions onBr(x),

supBr/2(x)

u ≤ C infBr/2(x)

u. (2)

In Sect. 5, we consider potentialsV satisfying the following decay ratecondition: There exists a constantC <∞ such that

V (x) ≤ C

1 + d(o, x)2(3)

for all x ∈M .For such a potentialV , the Harnack constantC appeared in(2) is indepen-dent ofR. Hence by adding the finite covering condition(FC), explainedin Sect. 5, on each end satisfying the conditions(V D)∞ and (P )∞, weget theL-regularity for the potentialV satisfying the decay rate condition(3). For example, these conditions(V D)∞, (P )∞ and(FC) are satisfiedon each end of a connected sum of complete Riemannian manifolds withnonnegative Ricci curvature.

An ingredient used in Sect. 5 is the concept of rough isometry, whichis a more general one than bi-Lipschitz map. We prove that the conditions(V D)∞, (P )∞ and (FC) are invariant under rough isometries betweencomplete Riemannian manifolds. But in deploying our theory via roughisometry, we need to add some local assumptions, explained in Sect. 5, oneachmanifold, due to the crudity of rough isometry. Consequently, if a com-plete Riemannian manifoldM is roughly isometric to a complete Rieman-nian manifold satisfying these conditions on each end, then theL-regularityfor a potentialV satisfying the decay rate condition(3) is valid on each endofM . Hence, we get the following result:

Theorem 4 LetM be a complete Riemannian manifold being roughly iso-metric to a complete Riemannian manifold with the volume doubling con-dition (V D)∞, the Poincare inequality(P )∞ and the finite covering con-dition (FC) on each end. LetL = ∆ − V be a Schrodinger operatoron M with a potentialV satisfying the decay rate condition(3). Sup-pose thatM has L-nonparabolic endsE1, E2, · · · , El, l ≥ 1, and L-parabolic endse1, e2, · · · , es, s ≥ 0. Then there exist positiveL-harmonicfunctionsfE1 , fE2 , · · · , fEl

, he1 , he2 , · · · , hes generating all positiveL-harmonic functions onM , i.e., for each positiveL-harmonic functionf ,


there exist positive constantsa1, a2, · · · , al andb1, b2, · · · , bs such that

f =l∑

i=1

aifEi +s∑

j=1

bjhej .

In particular, all boundedL-harmonic functions onM are generated byfE1 , fE2 , · · · , fEl

, i.e., for each boundedL-harmonic functionf , there existconstantsa1, a2, · · · , al such that

f =l∑

i=1

aifEi .


dimH+L (M) = s+ l, and dimHBL(M) = l.

Moreover, in the case thatM has onlyL-parabolic ends, every posi-tiveL-harmonic function and every boundedL-harmonic function must beidentically zero.

It is needed to point out that the number of ends is also invariant underrough isometries between complete Riemannian manifolds, moreover, therestriction map of a rough isometry to each end becomes a rough isometrybetween ends. (See Sect. 5.) As a corollary of Theorem 4, we have thefollowing new result for the Liouville theorem:

Corollary 1 (Liouville Theorem) LetM be a complete Riemannian man-ifold being roughly isometric to a complete Riemannian manifold with non-negative Ricci curvature. LetL = ∆ − V be a Schrodinger operator onM with a potentialV satisfying the decay rate condition(3). Then thespace of positive (bounded, respectively)L-harmonic functions is at mostone dimensional.

InSect. 6,we consider the spaceof positive harmonic functions ona classof manifolds, as an application of previous results. It has to be emphasizedthat the minimum principle holds in the case of the Laplacian, whereas thisis failed in the case of theSchrodinger operator. The validity of theminimumprinciple for harmonic functions enables us to state the regularity of endsunder more relaxed setting as follows: LetM be a complete Riemannianmanifold ando be a fixed point inM .We say that an endE ofM is regular ifthere exist a sequenceHr of compact (not necessarily connected) subsetsofE and a constantC <∞ such that for any nonnegative harmonic functionf onE,

supHr

f ≤ C infHr

f,


whered(o,Hr) → ∞ asr → ∞, and eachHr dividesE into a boundedsubsetAr and the unbounded componentUr of E \Hr.With this regularity, we have a similar result as that of Grigor’yan [5]. (SeeTheorem8.)Wealso consider ends satisfying the volumedoubling condition(V D)∞, the Poincare inequality(P )∞ and the finite covering condition(FC)′, explained in Sect. 6, which is a sufficient condition for regularityof each end. It is easy to prove that the parabolicity for the Laplacian isinvariant under rough isometries between ends. Thus we have the followingresults:

Theorem 5 LetM be a complete Riemannian manifold satisfying the vol-ume doubling condition(V D)∞, the Poincare inequality(P )∞ and thefinite covering condition(FC)′ on each end. LetN be a complete Rieman-nian manifold roughly isometric toM . Suppose thatM hasl nonparabolicends,l ≥ 1, ands parabolic ends,s ≥ 0, respectively. Then

dimH+(M) = s+ l = dimH+(N),

and

dimHB(M) = l = dimHB(N),

whereH+(X) andHB(X) denote the space spanned by positive harmonicfunctions and the space of bounded harmonic functions on the manifoldX,respectively.

Note that if a complete Riemannian manifoldM with nonnegative Riccicurvature outside a compact subset has finite first Betti number, thenMsatisfies the conditions(V D)∞, (P )∞ and(FC)′ at infinity of each end.(See [18] and [19].) Our result is more general than [17] and [18] in twodirections: One is that our assumption contains no curvature assumptionsand no topological restrictions like the finiteness of the first Betti number. Infact, since a rough isometry is not necessarily continuous, our result enablesone to expect that the behavior near infinity ofL-harmonic function dependsonmore crudemacroscopic analytic conditions rather than local geometrieslike curvatures or topological properties like the first Betti number. Thisresult partially answers the question of Li [16] related to the regularity ofends. The other is that the above analytic conditions are roughly isometricinvariant, and hence our result can be applied to a larger class of completeRiemannian manifolds. For example, Theorem 5 can be applied to the caseof a complete Riemannian manifold being roughly isometric to a completeRiemannian manifold with nonnegative Ricci curvature outside a compactset and finite first Betti number, which is a direct generalization of the resultof Li and Tam [18].


2 L-parabolicity and the Liouville function of ends

We begin with defining ends of a complete Riemannian manifoldM : Fixa pointo inM . We denote by%(r) the number of unbounded componentsofM \ Br(o). It is easy to prove that%(r) is nondecreasing inr > 0. Letlimr→∞ %(r) = k, wherek may be infinity, then we say that the numberof ends ofM is k. If k is finite, then we can chooser0 > 0 such that%(r) = k for all r ≥ r0. In this case, there exist mutually disjoint unboundedcomponentsE1, E2, · · · , Ek ofM \ Br0(o) and we call eachEi an end ofM for i = 1, 2, · · · , k. In particular, the number of ends is invariant underrough isometries between complete Riemannian manifolds. (See Sect. 5.)

We now classify ends of a complete Riemannian manifold by theL-parabolicity: LetL = ∆ − V be a Schrodinger operator on a completeRiemannian manifoldM . We say that an endE of M is L-nonparabolicif for somer1 ≥ r0, there exists a continuous functionuE , called anL-harmonic measure, onE \Br1(o) such that

LuE = 0 onE \Br1(o);uE = 0 on ∂Br1(o) ∩ E;

supE\Br1 (o) uE = 1.(4)

Otherwise, the endE is called anL-parabolic end. It is instructive to notethe result of Grigor’yan [6] that ifM is nonparabolic andV is a nonnegativefunction onM such that∫

MG(x, y)V (y)dy <∞,

whereG(x, y) is a positive Green’s function of the Laplacian∆, then thereexists a nontrivial boundedL-harmonic function onM . Furthermore, thisis equivalent to the existence of anL-harmonic functionu onM \ Ω suchthat0 ≤ u ≤ 1 onM \Ω, u = 0 on∂Ω, andsupM\Ω u = 1, whereΩ is aprecompact open subset ofM . (See [6].)

On the other hand, from theL-harmonicmeasureuE and the comparisonprinciple, we get a continuous functionvE of E \Br1(o) such that

L vE = 0 onE \Br1(o);vE = 1 on ∂Br1(o) ∩ E;

infE\Br1 (o) vE = 0,(5)

which plays a role as a barrier in Sect. 4.We now construct anL-harmonic functionLE on anL-nonparabolic

endE, which plays a crucial role in controlling the behavior near infinity ofboundedL-harmonic functionsonM . TheL-harmonic functionLE is called


the Liouville function ofE. Let LE,rr>r1 be a sequence of continuousfunctions on(Br(o) \Br1(o)) ∩ E such that

LLE,r = 0 on (Br(o) \Br1(o)) ∩ E;LE,r = 1 on ∂Br(o) ∩ E;LE,r = 1 on ∂Br1(o) ∩ E.

(6)

Then by the comparison principle,uE ≤ LE,r ≤ 1 on(Br(o)\Br1(o))∩E.SinceLE,r ismonotone decreasing, there exists its limit functionLE suchthatLLE = 0 onE anduE ≤ LE ≤ 1 onE. Furthermore,supE LE = 1andinfE(LE − uE) = 0, sincesupE uE = 1.

Next, we consider anL-harmonic functionL, called the Liouville func-tion of the whole manifoldM , which was introduced in [6] and [9]: LetLr be a sequence of continuous functions onM such that LLr = 0 on Br(o);

Lr = 1 on M \Br(o).

By the comparison principle,uE ≤ Lr ≤ LE,r on (Br(o) \ Br1(o)) ∩E for eachL-nonparabolic endE, whereuE is theL-harmonic measuregiven in (4) andLE,r is given in (6). Since the sequenceLr is monotonedecreasing, there exists its limit function, sayL. In particular,L is anL-harmonic function onM such thatuE ≤ L ≤ LE on eachL-nonparabolicendE. SincesupE uE = 1, we havesupE L = 1, infE(LE − L) = 0 andinfE(L− uE) = 0.

3 L-regularity and asymptotic behavior of solutions on ends

Wenow introduce theL-regularity of ends: LetL = ∆−V be aSchrodingeroperator on a complete RiemannianmanifoldM . Fix a pointo inM . We saythat an endE ofM isL-regular if there exists a constantC <∞ such thatfor sufficiently larger > 0 and for any nonnegativeL-harmonic functionfon (B2r(o) \Br/2(o)) ∩ E,

sup∂Br(o)∩E

f ≤ C inf∂Br(o)∩E

f.

We also assume the local Harnack inequality as follows: Letf be anonnegativeL-harmonic function onΩ andΩ′ be a relatively compactsubset ofΩ. Then there exists a constantC <∞ such that

supΩ′f ≤ C inf

Ω′ f,

whereC depends only onΩ andΩ′.


Throughout this section, each end is assumed to beL-regular unlessotherwise specified. TheL-regularity plays a crucial role in controlling thebehavior near infinity of positiveL-harmonic functions of each end. Thefollowing lemma is a key argument in analyzing the behavior near infinityof positiveL-harmonic functions:Lemma 1 LetE be an end of a complete Riemannian manifold andf be anonnegativeL-harmonic function onE.

(i) If lim supx→∞,x∈E

f(x) = ∞, then limx→∞,x∈E

f(x) = ∞.(ii) If lim inf

x→∞,x∈Ef(x) = 0, then lim

x→∞,x∈Ef(x) = 0.

Proof. We prove the first statement by the contradiction argument. Other-wise, we may assume that there exist a constantC1 < ∞ and a sequencexii∈N of points inE such thatd(o, xi) → ∞ asi→ ∞ and

f(xi) ≤ C1.

Thenwe can choose a sequencerii∈N of radii such thatxi ∈ ∂Bri(o)∩E.From theL-regularity ofE, we get

sup∂Bri (o)∩E

f ≤ CC1 and sup∂Bri+1 (o)∩E

f ≤ CC1

for eachi ∈ N. By the maximum principle on(Bri+1(o) \Bri(o)) ∩E, wehavef(x) ≤ CC1 for all x ∈ (Bri+1(o) \ Bri(o)) ∩ E. This implies thatf ≤ CC1 onE. This is a contradiction.

From the hypothesis of the second statement, we can choose a sequenceyii∈N of points inE such thatlimyi→∞ f(yi) = 0. Let rii∈N be asequence of radii satisfyingyi ∈ ∂Bri(o) ∩E. Then by theL-regularity ofE, we have

sup∂Bri (o)∩E

f ≤ Cf(yi)

for eachi ∈ N. Applying the maximum principle,

f(x) ≤ Cmaxf(yi), f(yi+1)

for all x ∈ (Bri+1(o) \Bri(o)) ∩ E. Hence we get

limy→∞,y∈E

f(y) = 0.


Applying the second result of Lemma 1 to theL-harmonic functionvEgiven in (5), we get

limx→∞,x∈E

vE(x) = 0. (7)

Similarly arguing, we can prove that if an endE is L-nonparabolic andL-regular, then both the Liouville functionLE of E and theL-harmonicmeasureuE of E have the same behavior near infinity ofE as follows:

Corollary 2 LetE be anL-nonparabolic andL-regular end of a completeRiemannian manifold. Then

limx→∞,∈E

(LE − uE)(x) = 0,

whereLE is the Liouville function ofE anduE is theL-harmonic measureofE defined in (6) and (4), respectively.

As a consequence of Lemma 1, we also have

limx→∞,x∈E

(LE − L)(x) = 0 and limx→∞,x∈E

(L− uE)(x) = 0.

The following lemma gives a characteristic of theL-nonparabolicity ofends:

Lemma 2 LetE be anL-nonparabolic end of a complete Riemannian man-ifold andf be a nonnegativeL-harmonic function onE. Thenf is boundedat infinity, i.e., there exists a constantCf <∞ such that

lim supx→∞,x∈E

f(x) ≤ Cf .

Proof. Otherwise, by Lemma 1, we may assume that

limx→∞,x∈E

f(x) = ∞.

Therefore, we can choose a constantA > 0 such thatf(y)/A < uE(y) forsomey ∈ E, whereuE is theL-harmonicmeasure ofE. Sincelimx→∞,x∈E

f(x) = ∞, by the comparison principle, we havef(x)/A ≥ uE(x) for allx ∈ E. This is a contradiction.

We also have a characteristic of anL-parabolic end as follows:Lemma 3 Let e be anL-parabolic (not necessarilyL-regular) end of acomplete Riemannian manifold andf be a nontrivialL-harmonic functionbounded above one \Br0(o) for somer0 > 0. Then

supef > lim sup

x→∞,x∈ef(x).


Proof. Suppose that

lim supx→∞,x∈e

f(x) = supef = m.

Sincef is nontrivial, there exists a proper open subsetΩ of e \Br0(o) suchthat

Ω = x ∈ e \Br0(o) : f(x) > m− εfor sufficiently smallε > 0. Put v = max(f − m + ε)/ε, 0. Thenvis a nonnegativeL-subharmonic function one \ Br0(o), v ≡ 0 on e \ Ω,andsupe\Br0 (o) v = 1. By the comparison principle, we can construct anL-harmonic functionue one \Br0(o) such that

Lue = 0 on e \Br0(o);ue = 0 on ∂Br0(o) ∩ e;

supe\Br0 (o) ue = 1.

This is a contradiction to theL-parabolicity of the ende.

4 Generalized Liouville theorem for the Schrodinger operator

Recall the result of Grigor’yan [6], in which he proved the validity of theLiouville theorem for the Schrodinger operator on parabolic manifolds asfollows: LetM be a parabolic complete Riemannian manifold, that is,Madmits no positive Green’s function for the Laplacian. Then every boundedsolution for the Schrodinger operatorL = ∆− V must be zero, wheneverthe nonnegative potentialV is not identically zero.

It has to be emphasized that in our setting, the Liouville theorem isderived from theL-parabolicity rather than the parabolicity for the Lapla-cian. Combining Lemma 3 and the maximum principle, we have the strongLiouville theorem (Theorem 2) for the Schrodinger operator.

Let L = ∆ − V be a Schrodinger operator on a complete RiemannianmanifoldM . From now on, each end ofM is assumed to beL-regularunless otherwise specified. We now construct generators, each of whichcorresponds to anL-nonparabolic end ofM , of the space of all boundedL-harmonic functions ofM .

Lemma 4 Let E be anL-nonparabolic end ofM . Then there exists aboundedL-harmonic functionfE onM such that

(i) L fE = 0 on M ;(ii) lim

x→∞,x∈E′ fE(x) = 0 for any otherL-nonparabolic endE′ ofM ;

(iii) limx→∞,x∈E

(fE − LE)(x) = 0,


whereLE is the Liouville function ofE given in (6).

Proof. Choose a sequencefr of continuous functions onBr(o) such that

Lfr = 0 on Br(o);fr = LE on ∂Br(o) ∩ E;fr = 0 on ∂Br(o) \ E.

Then by the comparison principle, we haveuE ≤ fr ≤ LE on Br(o) ∩ E;

0 ≤ fr ≤ vE′ on Br(o) ∩ E′,

whereE′ is any otherL-nonparabolic end ofM andvE′ is theL-harmonicfunction onE′ given in (5). The sequencefr is uniformly bounded, hencethere exists a subsequence converging uniformly to a limit functionfE onany compact subset ofM . In fact,fE is anL-harmonic function onM suchthat0 ≤ fE ≤ 1 onM and

uE ≤ fE ≤ LE on E;0 ≤ fE ≤ vE′ on E′.

From Corollary 2 and (7), we get the consequence. We stand in a position to describe the structure of the space of bounded

L-harmonic functions onM :

Theorem 6 Let E1, E2, · · · , El, l ≥ 1, be L-nonparabolic ends ofM .Then positiveL-harmonic functionsfE1 , fE2 , · · · , fEl

, each of which isgiven in Lemma 4, generate all boundedL-harmonic functions onM , i.e.,for each boundedL-harmonic functionf onM , there exist real numbersa1, a2, · · · , al such that

f =l∑

i=1

aifEi . (8)

Furthermore,fEi : i = 1, 2, · · · , l is linearly independent, hence

dimHBL(M) = l.

Proof. Let f be a boundedL-harmonic function onM . By Lemma 3, thereexists anL-nonparabolic endE such that

supEf = sup

Mf.

We may assume thatsupM f ≥ 0. By using the comparison principle andthe argument of construction ofLE , one can prove thatf ≤ CfLE ≤ Cf


onE, whereCf = supM f . SincesupE f = Cf , there exists a sequencexii∈N in E such that

limi→∞

(CfLE − f)(xi) = 0.

By Lemma 1, we have

limx→∞,x∈E

(CfLE − f)(x) = 0,

hence by Lemma 4,

limx→∞,x∈E

(CffE − f)(x) = 0.

Note thatf−CffE is also a boundedL-harmonic function onM satisfying

L (f − CffE) = 0 on M ;lim

x→∞,x∈E(f − CffE)(x) = 0;

limx→∞,x∈E′ (f − CffE) − f(x) = 0,

whereE′ is any otherL-nonparabolic end ofM . Repeating the above pro-cess, we can choose nonnegative real numbersc1, c2, · · · , cl such that

limx→∞,x∈E

(f −

l∑i=1

cifEi

)(x) = 0, (9)

wheneverE is anL-nonparabolic end satisfyinglim supx→∞,x∈E

f ≥ 0. Thus by

Lemma 3, we have

f −l∑

i=1

cifEi ≤ 0

onM . Putg =l∑

i=1cifEi −f . Theng is a nonnegative boundedL-harmonic

function onM . Applying the above argument tog again, there exist non-negative real numbersb1, b2, · · · , bl such that

limx→∞,x∈E

(g −

l∑i=1

bifEi

)(x) = 0, (10)

wheneverE is anL-nonparabolic end satisfyinglim infx→∞,x∈E

f ≤ 0. Combin-

ing (9) and (10), we have

f =l∑

i=1

aifEi ,


whereai = ci − bi for i = 1, 2, · · · , l.It is easy to prove that the setfEi : i = 1, 2, · · · , l is linearly indepen-

dent. Therefore, we have the consequence. According to the previous result,L-parabolic ends have no effects on

deciding the dimension of the space of boundedL-harmonic functions ona complete Riemannian manifold. But in the case of positiveL-harmonicfunctions, we are faced with quite different phenomenon with the case ofboundedL-harmonic functions.

The following lemma gives the existence of a positiveL-harmonic func-tion on anL-parabolic end, which plays a similar role to that ofL-harmonicmeasure in the case of anL-nonparabolic end:Lemma 5 Let e be anL-parabolic end ofM . There exists anL-harmonicfunctionve one \Br0(o) such that

(i) ve ≥ 0 one \Br0(o);(ii) ve = 0 on∂Br0(o) ∩ e;(iii) lim

x→∞,x∈eve(x) = ∞.

Proof. Consider a sequenceur of nonnegative continuous functions onesuch that

Lur = 0 on Br(o) ∩ e;ur = 0 on ∂Br0(o) ∩ e;ur = 1 on e \Br(o).

Since the sequenceur is monotone decreasing, there exists its limit func-tion u such thatLu = 0 on e and u = 0 on ∂Br0(o) ∩ e. By theL-parabolicity of the ende, the limit functionu must be identically zero. PutCr = sup∂Br1 (o)∩e ur for some fixedr1 > r0, thenCr → 0 asr → ∞.Define a new functionvr = ur/Cr onBr(o) ∩ e, thenvr satisfies

L vr = 0 on Br(o) ∩ e;vr = 0 on ∂Br0(o) ∩ e;vr = 1/Cr on e \Br(o).

LetK be a compact subset ofe containing∂Br1(o) ∩ e. Then by the localHarnack inequality, there exists a constantCK <∞ such that

supKvr ≤ CK inf

Kvr ≤ CK

for sufficiently larger > r1. This implies that the sequencevr is locallyuniformly bounded one. Hence, there exists a subsequence converging uni-formly to a limit functionve on any compact subset ofe. On the other hand,sincesup∂Br1 (o)∩e vr = 1 for sufficiently larger > r1, the limit functionveis not identically zero. From theL-parabolicity ofe, ve must be unbounded.Hence by Lemma 1, we get the consequence.


We now construct generators, each of which corresponds to eachL-parabolic end, of the space spanned by all positiveL-harmonic functions.EachL-harmonic functionve, introduced in Lemma 5, plays a role such as abarrier in constructing each generator corresponding to anL-parabolic ende as follows:

Lemma 6 Suppose thatM has at least oneL-nonparabolic end. Lete beanL-parabolic end ofM . Then there exists a positiveL-harmonic functionhe onM such that

(i) limx→∞,x∈e

he(x) = ∞;

(ii) limx→∞,x∈E

he(x) = 0 for anyL-nonparabolic endE ofM ;

(iii) 0 ≤ he ≤ a <∞ onM \ e.Moreover,h is unique up to a positive scalar multiple.

Proof. By Lemma 5, there exists a nonnegative continuous functionve one \Br0(o) such that

L ve = 0 on e \Br0(o);ve = 0 on ∂Br0(o) ∩ e;

limx→∞,x∈e

ve(x) = ∞.

Let hr be a continuous function onBr(o) such that

Lhr = 0 on Br(o);hr = ve on ∂Br(o) ∩ e;hr = 0 on ∂Br(o) \ e.

We first prove thathr is uniformly bounded on∂Br0(o) ∩ e. Otherwise,there exists a subsequenceri such thatari = max

∂Br0 (o)∩ehri → ∞ as

i→ ∞. Resetri = r and putHr = hr/ar onBr(o), then we have

Hr = ve/ar on ∂Br(o) ∩ e;Hr = 0 on ∂Br(o) \ e;

max∂Br0 (o)∩e

Hr = 1.

By the comparison principle, we get

0 ≤ Hr ≤ ve/ar + 1 on Br(o).

Thus the sequenceHr is locally uniformly bounded onM . Hence thereexists a subsequence converging uniformly to a limit functionH on anycompact subset ofM such thatLH = 0 onM and0 ≤ H ≤ 1 onM .


On the other hand, since0 ≤ Hr ≤ 1 onBr(o) \ (e \Br0(o)), we have0 ≤ Hr ≤ vE on Br(o) ∩ E, hence

limx→∞,x∈E

H(x) = 0,

whereE is anL-nonparabolic end ofM andvE is theL-harmonic functiongiven in (5). Applying Lemma 3 to the nonnegative boundedL-harmonicfunctionH, we haveH ≡ 0 onM . But choosing suitably a subsequence ofHr, we getmax∂Br0 (o)∩eH = 1, sincemax∂Br0 (o)∩eHr = 1. This is acontradiction. Consequently,hr is uniformly bounded on∂Br0(o) ∩ e.

Wenowconsider thebehavior ofhe at infinity ofM . Puttinga = supr ar,by the comparison principle,

ve ≤ hr ≤ ve + a on (Br(o) \Br0(o)) ∩ e.Thus we get

ve ≤ he ≤ ve + a on e \Br0(o),

hence (i). On the other hand, sincehr = 0 on ∂Br(o) \ e andhr ≤ a on∂Br0(o) ∩ e, by the comparison principle, we have

0 ≤ hr ≤ avE on Br(o) ∩ Efor eachL-nonparabolic endE. This implies that

0 ≤ he ≤ avE on E.

Henceweget (ii). Note thathr = 0on∂Br(o)\eandhr ≤ aon∂Br0(o)∩e.Therefore, by the maximum principle,

0 ≤ hr ≤ a on Br(o) \ (e \Br0(o)),

and hence (iii) immediately follows from

0 ≤ he ≤ a on M \ (e \Br0(o)).

To prove the uniqueness, leth be any other positiveL-harmonic functiononM such thatlimx→∞,x∈E h(x) = 0 for any L-nonparabolic endE,limx→∞,x∈e h(x) = ∞ andh is bounded on any otherL-parabolic end.

We claim thath/he ≤ c or he/h ≤ c for somec <∞. Otherwise, thereexist sequencesxi andyj in e such that

(h/he)(xi) → ∞ as d(o, xi) → ∞;(h/he)(yj) → 0 as d(o, yj) → ∞.


By theL-regularity ofe, we have

limri→∞

inf∂Bri (o)∩e h

sup∂Bri (o)∩e he→ ∞ and lim

tj→∞

sup∂Btj (o)∩e h

inf∂Btj (o)∩e he→ 0,

whereri = d(o, xi) and tj = d(o, yj). Thus we can choose a positiveconstantc > 0 and sufficiently large0 < r1 < t2 < r3 <∞ such that

inf∂Br1 (o)∩e

h ≥ c sup∂Br1 (o)∩e

he, inf∂Br3 (o)∩e

h ≥ c sup∂Br3 (o)∩e

he (11)

andsup

∂Bt2 (o)∩eh ≤ (c/2) inf

∂Bt2 (o)∩ehe. (12)

By the comparison principle, (11) implies that

h ≥ che on (Br3(o) \Br1(o)) ∩ e. (13)

Applying (13) to (12), there exists a pointx ∈ ∂Bt2(o) ∩ e such thath(x) ≥ che(x) ≥ 2h(x).

This is a contradiction to the fact thath is positive one. Therefore, we havethe claim.

We now assume thatch ≥ he on e. Note thatch − he is bounded onany otherL-parabolic ends, andlimx→∞,x∈E (ch − he)(x) = 0 for anyL-nonparabolic endE ofM . Therefore, applying Lemma 3 tohe − ch, wehavech− he ≥ 0 onM .

Setc = infc : ch ≥ he, then0 < c < ∞ becausehe > 0. We claimthat there exists a constantc > 0 such thathe = ch onM . In order to provethe claim, we have only to consider the case thatch−he ≥ 0 is unboundedone. Otherwise, the claim immediately follows from Lemma 3 and the factthatlimx→∞,x∈E(ch−he)(x) = 0 for eachL-nonparabolic endE. Arguingsimilarly as above, there exists a constantd such that

he

ch− he≤ d or

ch− hehe

≤ d.

If he/(ch − he) ≤ d, then(d + 1)he ≤ dch. This is a contradiction to thedefinition of the constantc. If (ch− he)/he ≤ d, thench ≤ (d+ 1)he. Setd = supd : dh ≤ he. By the local Harnack inequality, we may assumech−he > 0 andhe −dh > 0, each of which is unbounded one. Thus thereexistc1 > 0 andc2 > 0 such that

ch− he ≥ c1(he − dh) or ch− he ≤ c2(he − dh),


i.e.,c+ c1d1 + c1

h ≥ he orc+ c2d1 + c2

h ≤ he.

Each of them contradicts the definitions ofc andd, respectively. Thus thereexists a constantc > 0 such thathe = ch. The remains are similar.

Combining Theorem 6 and Lemma 6, we can prove the rest part ofTheorem 3 as follows:

Proof of Theorem 3Let f be a positiveL-harmonic function onM . Thenby Lemma 2, there exists a nonnegative constantCEi < ∞ for eachi =1, 2, · · · , l such that

lim infx→∞,x∈E

f(x) = CEi .

By the comparison principle,

f ≥ CEiuEi on Ei,

whereuEi is theL-harmonicmeasure onEi given in (4). Hence there exist anonnegative constantdi and a sequencexn inEi such thatf −diuEi > 0onEi and(f − diuEi)(xn) → 0 asd(o, xn) → ∞. By Lemma 1, we have

limx→∞,x∈Ei

(f − diuEi)(x) = 0,

hencelim

x→∞,x∈Ei

(f − difEi)(x) = 0.

Repeating this process, we get nonnegative constantsd1, d2, · · · , dl suchthat

limx→∞,x∈Ek

(f −

l∑i=1

difEi

)(x) = 0,

for eachk = 1, 2, · · · , l. If f−∑li=1 difEi is bounded on everyL-parabolic

end, then by Lemma 3,

f =l∑

i=1

difEi on M.

Assume thath = f −∑li=1 difEi is unbounded onL-parabolic ends

e1, e2, · · · , et for some1 ≤ t ≤ s. We claim that for eachj = 1, 2, · · · , t,there exists a constant0 < cj < ∞ such thath − cjhej is bounded onej .Otherwise, there exists anL-parabolic endej for some1 ≤ j ≤ t such thatfor any constant0 < c <∞,

h− chej is still unbounded onej . (14)


Similarly as in the proof of Lemma 6, there exists a constant0 < cj < ∞such that

h ≥ cjhej or h ≤ cjhej on ej . (15)

First, assume thatcjhej −h ≥ 0 onei. Setci = infcj : cjhej ≥ h on ej,then0 < cj <∞ becauseh > 0. By (14),cjhej −h ≥ 0 is still unboundedonej . On the other hand, by (15) and the definition of the numbercj , thereexists a constant0 < aj <∞ such that

(cj − aj)hej ≤ h.

Setcj = supcj : cjhej ≤ h on ej. If h = cjhej , then we get the claim.Otherwise, by the local Harnack inequality, we may assume that

cjhej − h > 0 and h− cjhej > 0 on ej , (16)

and each of them is unbounded onej . Thus there exists a constant0 < bj <∞ such that

cjhej − h ≥ bj(h− cjhej ) or cjhej − h ≤ bj(h− cjhej ) on ej ,

hencecj + bjcjbj + 1

hej ≥ h orcj + bjcjbj + 1

hej ≤ h on ej .

From thedefinitionofcj andcj ,wehavecj = cj . This implies thatcjhej = hor cjhej = h onej . This is a contradiction to (16).

Consequently, for eachj = 1, 2, · · · , t, there exists a constant0 < cj <∞ such thath − cjhj is bounded onej . For such constants0 < cj < ∞,h−∑t

j=1 cjhej is bounded onM and

limx→∞,x∈Ei

h−

t∑j=1

cjhej

(x) = 0

for all i = 1, 2, · · · , l. By Lemma 3, we haveh =∑t

j=1 cjhej onM , i.e.,

f =l∑

i=1

difEi +t∑

j=1

cjhej on M.

It is easy to prove that the setfEi , hej : i = 1, 2, · · · , l, j = 1, 2, · · · , sis linearly independent.


5 L-regularity and rough isometric invariants

In previous sections, we used the concept of theL-regularity of ends, whichasserts the validity of the sphere version of the Harnack inequality near in-finity of each end for nonnegativeL-harmonic functions. In this section, wetreat the Schrodinger operatorL = ∆−V on a complete Riemannian man-ifoldM with a potentialV satisfying the decay rate condition (3). As men-tioned in Sect. 1, we get the Harnack inequality for nonnegative solutionsfor the Schrodinger operatorL on each ball near infinity of ends satisfyingthe volume doubling condition(V D)∞ and the Poincare inequality(P )∞as follows: LetE be an end ofM . Fix a pointo inM .

(V D)∞ for given0 < α < 1/2, there is a constantC <∞ depending onlyonαsuch that for anypointx ∈ ∂BR(o)∩E andany0 < r < R/2,

volBr(x) ≤ CvolBαr(x),

whereR is sufficiently large;(P )∞ there exist a constantC < ∞ and an integerk ∈ N such that

for any pointx ∈ ∂BR(o) ∩ E, any0 < r < R/2 and allf ∈C∞(Br(x)),∫

Br/k(x)|f − f |2 ≤ Cr2

∫Br(x)

|∇f |2,

wheref = (volBr/k(x))−1 ∫Br/k(x) f andR is sufficiently large.

It is well known that these conditions hold on any complete Riemannianmanifoldwith nonnegativeRicci curvature outside a compact set. (See [20].)Even now, in order to obtain the validity of theL-regularity of ends, we needsome uniform connectedness near infinity of each end. We now introducethe following finite covering condition:

(FC) for given0 < α < 1/4, there exist an integerm = m(α) and pointsx1, x2, · · · , xm in ∂BR(o)∩E such that for sufficiently largeR > 0

∂BR(o) ∩ E ⊂m⋃i=1

BαR(xi)

and⋃m

i=1BαR(xi) is connected.

One may be somewhat unfamiliar with the finite covering condition(FC). But the condition(FC) holds on a connected sum of complete Rie-mannian manifolds with nonnegative Ricci curvature. (See [1] and [19].)

From the conditions(V D)∞, (P )∞ and(FC), we get theL-regularityof ends as follows: LetE be an end ofM satisfying the conditions(V D)∞,


(P )∞ and(FC). Then there exists a constantC < ∞ such that for anynonnegativeL-harmonic functionf onE and for sufficiently largeR > 0,

sup∂BR(o)∩E

f ≤ C inf∂BR(o)∩E

f.

Now, the programs in Sect. 3 and Sect. 4 can be applied to this situation.Hence for such a manifold, we get Theorem 3.

It is a natural question to ask how wider class of manifolds Theorem3 can be extended to. In the rest of this section, we prove that if an endE is roughly isometric to an end satisfying the conditions(V D)∞, (P )∞and (FC), then theL-regularity for the potentialV satisfying the decayrate condition(3) is also valid onE. As a consequence of this result, weobtain that if a complete Riemannian manifoldM is roughly isometric toa complete Riemannian manifold having finitely many ends satisfying theconditions(V D)∞, (P )∞ and(FC), then the dimension of the space ofpositive (bounded, respectively)L-harmonic functions onM is the same asthe number of ends (L-nonparabolic ends, respectively).

We begin with introducing some notions and some local assumptions:A map, not necessarily continuous,ϕ : X → Y is called a rough isometry,introduced by Kanai in [11], [12] and [13], between two metric spacesXandY if ϕ satisfies the following conditions:

(R1) for someτ > 0, theτ -neighborhood of the imageϕ(X) coversY ;(R2) there exist constantsa ≥ 1 andb ≥ 0 such that

1ad(x1, x2) − b ≤ d(ϕ(x1), ϕ(x2)) ≤ ad(x1, x2) + b

for all x1, x2 ∈ X, whered denotes the distances ofX andY inducedfrom their metrics, respectively.

From now on,τ, a andb alwaysmean those which appear in(R1) and(R2).Especially, being roughly isometric is an equivalence relation. (See [12].)It is also important to note that two roughly isometric metric spaces mayhave completely different topology, since the rough isometry is not assumedto be continuous. For example, an infinite cylinder is roughly isometric toan infinite cylinder with infinitely many identical handles attached at equaldistance going off to the infinity. So, in order to deploy our theory viarough isometry, it is needed to add the following local conditions on eachRiemannian manifold: Letϕ : M → N be a rough isometry betweencomplete Riemannian manifoldsM andN .

(V C)loc there exists a constantC ≥ 1 such that for any pointx inM

1C

volB1(x) ≤ volB1(ϕ(x)) ≤ CvolB1(x);


(V D)loc there exists a constantCr <∞ depending only onr such that forany pointx inM (in N , respectively)

volB2r(x) ≤ CrvolBr(x);

(P )loc there exists a constantCr < ∞ depending only onr > 0 suchthat for each pointx in M (in N, respectively) and for allf ∈C∞(Br(x)) ∫

Br(x)|∇f | ≥ Cr

∫Br(x)

|f − f |,

wheref = (volBr(x))−1 ∫

Br(x) f .

These local assumptions are satisfied on any completeRiemannianmanifoldwithRicci curvature boundedbelowbya constant and thepositive injectivityradius. (See [2], [12] or [4].) Fromnowon,whenwesay that amapϕ :M →N is a rough isometry between complete Riemannian manifoldsM andN ,it means that the complete RiemannianmanifoldsM andN satisfy the localconditions(V D)loc and(P )loc, and themapϕ satisfies the conditions(R1),(R2) and(V C)loc, unless otherwise specified.

We first prove that the number of ends of a complete Riemannian man-ifold is invariant under rough isometries between complete Riemannianmanifolds, moreover, the restriction map of a rough isometry to each endbecomes a rough isometry between ends as follows:

Lemma 7 Letϕ :M → N be a rough isometry(not necessarily satisfyingthe condition(V C)loc) between complete Riemannian manifoldsM andN (not necessarily satisfying the conditions(V D)loc and(P )loc). SupposethatM has endsD1, D2, · · · , Dk. ThenN has alsok endsE1, E2, · · · , Ek.Moreover, the rough isometryϕ induces a rough isometry between ends insuch a way that for eachi = 1, 2, · · · , k the restriction mapϕ|Di : Di → Ei

is also a rough isometry,ϕ(Di) ⊂ Ei andBτ (ϕ(Di)) ⊃ Ei \ Bt0(o′) for

somet0 > 0, whereo′ = ϕ(o) for a fixed pointo inM .

Proof. Choose a numberr0 > 0 such thatD1, D2, · · · , Dk are mutuallydisjoint unbounded components ofM \ Br0(o) and a constantc > 0 suchthat all bounded components ofM \Br0(o) are contained inBr0+c(o).

First, we claim that the number of ends ofN is less than or equal tok.Otherwise, there exists a constantt0 > a(r0+c)+b such that the number ofunbounded components ofN \Bt0(o

′) is greater thank. So wemay assumethat there exist mutually disjoint unbounded componentsE1, E2, · · · , Ek+1ofN \Bt0(o

′) such thatBτ (ϕ(D1∪D2∪· · ·∪Dk)) ⊃ E1∪E2∪· · ·∪Ek+1.Then by the definition of the rough isometry, there exist an endDi and asequencexnn∈N in Di satisfying:


(i) d(xn, xn+1) = 1 for eachn ∈ N;(ii) lim

n→∞ d(ϕ(yn), o′) = ∞, whereyn is a subsequence ofxn such

thatϕ(yn) ∈ Ej for somej = 1, 2, · · · , k + 1;(iii) lim

n→∞ d(ϕ(zn), o′) = ∞, wherezn is the complement ofyn in

xn, henceϕ(zn) ∈ (N \Bt0(o′)) \ Ej .

Now we can choose a sufficiently largen0 ∈ N such that

ϕ(xn0) ∈ Ej and ϕ(xn0+1) ∈ (N \Bt0(o′)) \ Ej .

Thus for such ann0 ∈ N, we have

d(ϕ(xn0), Bt0(o′)) > a+ b and d(ϕ(xn0+1), Bt0(o

′)) > a+ b.

Sinced(ϕ(xn0), ϕ(xn0+1)) ≤ a+ b, this is a contradiction. Consequently,the number of ends ofN is less than or equal tok. On the other hand, sincethere exists a rough isometryψ : N → M , the number of ends ofN mustbek.

From the above result, we can assign eachDi to Eji such thatϕ(Di \Br0(o)) ⊂ Eji for somer0 > 0. We have only to prove that ifϕ(Di1 \Br0(o)) ⊂ Eji1

andϕ(Di2 \ Br0(o)) ⊂ Eji2for i1 = i2, thenji1 = ji2 .

Otherwise,ϕ((Di1 ∪Di2) \ Br0(o)) ⊂ Eji1. Then there exists a sequence

xnn∈N in Eji1satisfying:

(i) d(xn, xn+1) = 1 for eachn ∈ N;(ii) lim

n→∞ d(yn, o′) = ∞, whereyn is a subsequence ofxn such thatyn ∈ Bτ (ϕ(Di1 \Br0(o)));

(iii) limn→∞ d(zn, o′) = ∞, wherezn is the complement ofyn in xn,hencezn ∈ Bτ (ϕ(Di2 \Br0(o))).

We can choose a sufficiently largen0 ∈ N such thatxn0 ∈ yn andxn0+1 ∈ zn+1. This is a contradiction tod(xn0 , xn0+1) = 1. Let usrearrangeji such thatji = i. So we have the consequence.

From the local conditions(V C)loc, (V D)loc and the definition of roughisometry, it is easy to prove that the volume doubling condition(V D)∞ isa rough isometric invariant between ends as follows:

Lemma 8 LetD andE be ends of complete Riemannian manifoldsM andN , respectively, andD satisfy the volume doubling condition(V D)∞. Letϕ : D → E be a rough isometry ando′ = ϕ(o) for a fixed pointo in M .Then for each0 < α < 1/2, there exists a constantC <∞ depending onlyonα such that

volBr(y) ≤ CvolB2a2αr(y),

wherey ∈ ∂BR(o′) ∩ E and0 < r < R/8a2.


Recently, Coulhon and Saloff-Coste [4] proved that the Poincare in-equality is invariant under rough isometries between complete Riemannianmanifolds. In our setting, since each ball must be contained in an end, theirresult is not fully available. But slightly modifying the program in [4], onecan prove the rough isometric invariance of the Poincare inequality(P )∞as follows:

Lemma 9 LetD andE be ends of complete Riemannian manifoldsM andN , respectively, andD satisfy thePoincare inequality(P )∞. Letϕ : D → Ebe a rough isometry ando′ = ϕ(o) for a fixed pointo in M . Then thereexist a constantC < ∞ and an integerk′ ∈ N such that for any pointy ∈ ∂BR(o′) ∩ E, any0 < r < R/2 and allf ∈ C∞(Br(y)),∫

Br/k′ (y)|f − f |2 ≤ Cr2

∫Br(y)

|∇f |2,

wheref = (volBr/k′(y))−1 ∫Br/k′ (y) f andR is sufficiently large.

By using the standard Moser iteration, together with Lemma 8 andLemma 9, one can achieve the Harnack inequality for the Schrodinger oper-atorL with potentialV satisfying the decay rate condition(3) on each ballnear infinity of the endE. (See [15] and [10].)

Now we prove that the finite covering condition(FC) is also invariantunder rough isometries between ends. In order to handle the condition(FC)via rough isometry, we need some modification of the condition(FC) asfollows:

Remark 1In (FC), since⋃m

i=1BαR(xi) is connected, wemay assume that

BαR(xi) ∩BαR(xi−1) = ∅

for eachi = 2, 3, · · · ,m. Therefore, by adding some balls centered on∂BR(o)∩D, we can rephrase the finite covering condition(FC) as follows:For given0 < α < 1/4 and for sufficiently largeR > 0, there is an integerm = m(α) andx1, x2, · · · , xm ∈ ∂BR(o)∩D such thatd(xi, xi−1) ≤ αRfor eachi = 2, 3, · · · , m,

BαR/2(∂BR(o) ∩D) ⊂m⋃i=1

BαR(xi), (17)

and⋃m

i=1BαR(xi) is connected.Choose a smallest integern ∈ N such thatn ≥ 8a2/α, and a finite

sequenceRj : j = 0, 1, 2, · · · , n such thatR0 = R/2a, Rj = Rj−1 +


αR/4a for j = 1, 2, · · · , n. By (17), there exist pointsxj1, xj2, · · · , xjm ∈

∂BRj (o) ∩D for eachj = 0, 1, 2, · · · , n such that

BαRj/2(∂BRj (o) ∩D) ⊂m⋃i=1

BαRj (xji )

and⋃m

i=1BαRj (xji ) is connected. Consequently, we have

(Ba(R+τ+b)(o) \B(R−τ−b)/a(o)) ∩D ⊂n⋃

j=0

m⋃i=1

BaαR(xji ) (18)

and⋃n

j=0⋃m

i=1BaαR(xji ) is connected.

Lemma 10 LetD andE be ends of complete Riemannian manifoldsMandN , respectively, andD satisfy the finite covering condition(FC). Letϕ : D → E be a rough isometry ando′ = ϕ(o) for a fixed pointo in M .Then

∂BR(o′) ∩ E ⊂n⋃

j=0

m⋃i=1

B3a2αR(ϕ(xji ))

and⋃n

j=0⋃m

i=1B3a2αR(ϕ(xji )) is connected, where the pointsxji ’s aregiven in(18).

Proof. Sinced(xji−1, xji ) ≤ αR for eachi = 2, 3, · · · , m, ⋃m

i=1B3aαR

×(ϕ(xji )) is connected. Note thatd(∂BRj−1(o), ∂BRj (o)) ≤ αR/2a forthe sequenceRjj=1,2,···,n given in Remark 1. Thus there exist pointswj ∈ ∂BRj (o)∩D such thatd(wj−1, wj) ≤ αR/2a. By the condition(R2),we getd(ϕ(wj−1), ϕ(wj)) ≤ αR/2+ b, hence

⋃nj=0⋃m

i=1B3a2αR(ϕ(xji ))is also connected. By the definition of the rough isometry, we have

∂BR(o′) ∩ E ⊂ Bτ (ϕ((Ba(R+τ+b)(o) \B(R−τ−b)/a(o)) ∩D))

⊂ Bτ

ϕ

n⋃

j=0

m⋃i=1

BαR(xji )

⊂n⋃

j=0

m⋃i=1

B3a2αR(ϕ(xji )).

Fromabove results,weget theL-regularity of eachend roughly isometric

to an end satisfying the conditions(V D)∞, (P )∞ and(FC) as follows:


Theorem 7 (Harnack Inequality) LetD andE be ends of complete Rie-mannian manifoldsM andN , respectively, andD satisfy the conditions(V D)∞, (P )∞ and (FC). Let ϕ : D → E be a rough isometry ando′ = ϕ(o) for a fixed pointo inM . LetL = ∆− V be a Schrodinger oper-ator onN with a potentialV satisfying the decay rate condition(3). Thenthere exist a constantC < ∞ such that for any nonnegativeL-harmonicfunctionf defined on the endE,

sup∂BR(o′)∩E

f ≤ C inf∂BR(o′)∩E

f.

Applying the programs of Sect. 3 and Sect. 4 to this situation, we haveTheorem 4.

By the splitting theorem of Cheeger and Gromoll [3], if a completeRiemannian manifoldM has nonnegative Ricci curvature, thenM has atmost two ends. In particular, ifM has two ends, then each end ofM mustbe parabolic for the Laplacian onM . From this fact, together with Theorem7, we have the following corollary:

Corollary 3 LetM be a connected sum of complete Riemannian manifoldswith nonnegativeRicci curvature. LetN beacompleteRiemannianmanifoldbeing roughly isometric toM andL = ∆ − V be a Schrodinger operatoronN with a potetialV satisfying the decay rate condition(3). If N has atleast oneL-nonparabolic end, then the dimension of the space of positiveL-harmonic functions is equal to the number of ends ofN , hence that ofM .Furthermore, the dimension of the space of boundedL-harmonic functionsis equal to the number ofL-nonparabolic ends ofN .

In particular, if N has onlyL-parabolic ends, then every positiveL-harmonic function and every boundedL-harmonic function onN must beidentically zero.

As a simpler application, we have Corollary 1.

6 Application to harmonic functions

In this section, we deal with solutions for the Laplacian operator as a trivialcase of the Schrodinger operator. It is easier to handle the behavior of har-monic functions than that ofL-harmonic functions. Asmentioned in Sect. 1,we get the validity of the minimum principle, which enables us to definethe regularity under more general setting as follows: LetM be a completeRiemannian manifold ando be a fixed point inM . We say that an endE ofM is regular if there exist a sequenceHr of compact (not necessarily con-nected) subsets ofE and a constantC < ∞ such that for any nonnegative


harmonic functionf onE,

supHr

f ≤ C infHr

f,

whered(o,Hr) → ∞ asr → ∞, and eachHr dividesE into a boundedsubsetAr and the unbounded componentUr of E \Hr.

We first prove that every positive harmonic function is asymptoticallyconstant on each end satisfying the regularity. Then we can achieve similarresults to those in Sect. 3 and Sect. 4. It is needed to note that the proof isslightly different because the setting of the regularity is more relaxed thanthat of theL-regularity, to be precise, each sphere∂Br(o) in theL-regularityis replaced byHr in the regularity. But this handicap can be overcome bythe minimum principle for harmonic functions.

Lemma 11 LetE be a regular end of a complete Riemannian manifoldMand f be a nonnegative harmonic function onE. Then there exists0 ≤Cf ≤ ∞ such that

limx→∞,x∈E

f(x) = Cf .

Proof. Suppose thatf is unbounded onE. We claim thatlimx→∞,x∈E f(x)= ∞. Otherwise, we may assume that there exist a constantC1 < ∞ anda sequencexii∈N of points inE such thatd(o, xi) → ∞ as i → ∞andf(xi) ≤ C1. Then there exists a sequencerii∈N of positive numberssuch thatxi ∈ Uri \Uri+1 , whereUri denotes the unbounded component ofE \Hri . By the minimum principle, we get

infHri

f ≤ C1,

whereri = ri or ri+1. From the regularity ofE,

supHri

f ≤ CC1.

Applying the maximum principle tof onUri \Uri+1 , we havef(x) ≤ CC1for all x ∈ Uri \ Uri+1 . This implies thatf ≤ CC1 onE. This contradictsthatf is unbounded onE.

Next, we prove the case thatf is bounded onE. Suppose that there existsa sequencexii∈N inE such thatlimxi→∞ f(xi) = 0. Choose a sequencerii∈N such thatxi ∈ Uri \ Uri+1 . Arguing similarly as above, we have

limx→∞,x∈E

f(x) = 0.

Since every constant is also a harmonic function, the remains immediatelyfollow from the above argument.


Substituting the role of the Liouville function in Sect. 4 with the constantfunction 1, we have the similar result to that of Grigor’yan [5] as follows:

Theorem 8 LetM be a complete Riemannian manifold with regular ends.Suppose thatM hasnonparabolicendsE1, E2, · · · , El, l ≥ 1, andparabolicendse1, e2, · · · , es, s ≥ 0. Then there exist positive harmonic functionsfE1 , fE2 , · · · , fEl

and he1 , he2 , · · · , hes generating all positive harmonicfunctions, i.e., for each positive harmonic functionf , there exist positiveconstantsa1, a2, · · · , al andb1, b2, · · · , bs such that

f =l∑

i=1

aifEi +s∑

j=1

bjhej .

In particular, all bounded harmonic functions are generated by positive har-monic functionsfE1 , fE2 , · · · , fEl

, i.e., for each bounded harmonic functionf , there exist constantsa1, a2, · · · , al such that

f =l∑

i=1

aifEi .


dimH+(M) = s+ l, and dimHB(M) = l.

Now we consider a sufficient condition of the regularity of ends. In thissituation, the uniform connectedness assumed in(FC) can be relaxed asfollows: Let E be an end of a complete Riemannian manifold andCE,R

denote the unbounded component ofE \BR(o).

(FC)′ for given0 < α < 1/4, there exist an integerm = m(α) and pointsx1, x2, · · · , xm in ∂CE,R such that for sufficiently largeR > 0

∂CE,R ⊂m⋃i=1

BαR(xi)

and⋃m

i=1BαR(xi) is connected.

It is not difficult to check that if an endE of a complete Riemannianmanifold satisfies the volume doubling condition(V D)∞, the Poincare in-equality(P )∞ and the finite covering condition(FC)′, thenE is a regularend.

As mentioned in Sect. 5, the volume condition(V D)∞ and the Poincareinequality(P )∞ are invariant under rough isometries between ends. Sincethe uniform connectedness assumed in the condition(FC)′ is weaker than


that of the condition(FC), it is not clear whether the finite covering con-dition (FC)′ itself is a roughly isometric invariant. Nonetheless, we obtaina modified version of the finite covering condition, however, which is suffi-cient to guarantee the regularity:

Lemma 12 LetE be an end roughly isometric to an endD satisfying thefinite covering condition(FC)′. Then there exist a sequenceHR of hy-persurfaces inE and a positive integerm such thatd(o′, HR) → ∞ asR→ ∞,

HR ⊂m⋃i=1

B3a2αR(ϕ(xi))

and⋃m

i=1B3a2αR(ϕ(xi)) is connected. In particular, each hypersurfaceHR

dividesE into two partsAR andUR, whereAR is a bounded subset andUR is the unbounded component ofE \HR, respectively.

Proof. Similarly as the argument in Remark 1, there exist pointsxj1, xj2, · · · ,

xjm in ∂CD,Rj such that

BαRj/2(∂CD,Rj ) ⊂m⋃i=1

BαRj (xji )

and⋃m

i=1BαRj (xji ) is connected,whereRj is a sequencegiven inRemark

1. Moreover, we get

(Ba(R+τ+b)(o) \B(R−τ−b)/a(o)) ∩D ⊂n⋃

j=0

m⋃i=1

BaαR(xji ) ∪n⋃

j=0

Aj

and⋃n

j=0⋃m

i=1BaαR(xji ) is connected, whereAj is the union of boundedcomponents ofE \ BRj (o). By the similar argument as in Remark 1,⋃n

j=0⋃m

i=1B3a2αR(ϕ(xji )) is connected.We now prove that there exists a sequenceHR of hypersurfaces sat-

isfying the above statement. Otherwise, we may assume that there exist apoint y ∈ ∂CE,R \ (

⋃nj=0⋃m

i=1B3a2αR(ϕ(xji ))), whereCE,R denotes theunbounded component ofE \BR(o′), and an arclength parametrized curveγ : [0,∞) → E \BR(o′) such thatγ(0) = y, γ(t) → ∞ ast→ ∞ and

Bc(γ[0,∞)) ∩ n⋃

j=0

m⋃i=1

B2a2αR(ϕ(xji ))

= ∅, (19)

wherec > 5a2(τ+a+b). By the conditions(R1) and(R2), we can choosea sequencezkk∈N in D \ BR/2a(o) such thatd(ϕ(zk), γ(k)) ≤ τ for


eachk ∈ N. Therefore, there exists a curveσ : [0,∞) → D \ BR/2a(o)such thatσ(0) ∈ ∂BR/2a(o) ∩ D, σ(t) → ∞ ast → ∞, andσ[0,∞) ⊂⋃∞

k=0Ba(2τ+b+1)(zk). Henceby the condition(R2), wehaveϕ(σ[0,∞)) ⊂B4a2(τ+a+b)(γ[0,∞)). This implies that

Bτ (ϕ(σ[0,∞))) ⊂ Bc(γ[0,∞)). (20)

On the other hand, sinceσ(0) ∈ B2aR(o) andσ(t) → ∞ ast → ∞,there exists a pointx0 ∈ σ[0,∞) ∩ ∂CD,2aR. Hence by (20),ϕ(x0) ∈Bc(γ[0,∞)). But since

ϕ(∂CD,2aR) ⊂ Bτ

ϕ

n⋃

j=0

m⋃i=1

BαR(xji )

⊂

n⋃j=0

m⋃i=1

B2a2αR(ϕ(xji )),

this is a contradiction to (19). Theorem 9 (Harnack Inequality) LetD andE be ends of complete Rie-mannian manifoldsM andN , respectively, andD satisfy the conditions(V D)∞, (P )∞ and(FC)′. Letϕ : D → E be a rough isometry. Then thereexists a constantC < ∞ such that for any nonnegative harmonic functionf on the endE,

supHR

f ≤ C infHR

f,

where eachHR is the hypersurface given in Lemma 12.

Substituting the role of∂CD,R in the proof of Lemma 11 withHR, wehave the same result on the roughly isometric class:

Lemma 13 Let anendE be roughly isometric toD satisfying the conditions(V D)∞, (P )∞ and(FC)′. Then for any nonnegative harmonic functionfonE, there exists0 ≤ Cf ≤ ∞ such that

limx→∞,x∈E

f(x) = Cf .

Before proving Theorem 5, it is needed to introduce the parabolicityof each end. All ends are divided into two classes by the parabolicity: LetM be a complete Riemannian manifold. We say that an endD of M isnonparabolic, if for somer0 > 0, there exists a continuous functionudefined onD \Br0(o) such that

∆u = 0 on D \Br0(o);u = 1 on ∂Br0(o) ∩D;

infD u = 0.


Otherwise, the endD is called a parabolic end. Now define the capacity ofeach endD ofM by

Cap(D \Br0(o)) = infu

∫D\Br0 (o)

|∇u|2,

where the infimum is taken over all continuous functionsu onD \ Br0(o)such thatu is smooth onD\Br0(o), vanishing at infinity andu|∂Br0 (o)∩D =1. It is easy to prove that the positivity ofCap(D \Br0(o)) is equivalent tothe nonparabolicity of the endD.

Slightly modifying Kanai’s program [13], one can prove that the pos-itivity of the capacity of each end is preserved under rough isometries asfollows:

Lemma 14 LetD andE be ends of complete Riemannian manifoldsMandN , respectively. Letϕ : D → E be a rough isometry. Suppose thatCap(D \ Br0(o)) > 0, then we haveCap(E \ BR0(o

′)) > 0 for someR0 ≥ 2(ar0 + b), whereo′ = ϕ(o) for a fixed pointo inM .

Proof of Theorem 5LetM be a complete Riemannian manifold satisfyingthe volume doubling condition(V D)∞, the Poincare inequality(P )∞ andthe finite covering condition(FC)′ on each end. LetN be a completeRiemannian manifold roughly isometric toM . By Theorem 9, each endof N becomes a regular end. Furthermore, by Lemma 14, the parabolicityof ends is preserved under rough isometries between ends. Therefore, byTheorem 8, we have Theorem 5.

It is important to note that the finite covering condition(FC)′ holds ona complete Riemannian manifold with nonnegative Ricci curvature outsidea compact set and finite first Betti number. (See [19] and [18].)

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Generalized Liouville property for Schrödinger operator on Riemannian manifolds

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