The Rigidity of Ricci shrinkers of dimension four

0.23 MB·

by  Yu Li

The rigidity of Ricci shrinkers of dimension four
Yu Li and Bing Wang
∗
February 21,2017
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Abstract
e
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Indimension4,weshowthatanontrivialflatconecannotbeapproximatedbysmoothRicci

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shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-
1
Hausdorfftopology.Asapplications,weobtainuniformpositivelowerboundsofscalarcurva-

] tureandpotentialfunctionsonRiccishrinkerssatisfyingsomenaturalgeometricproperties.
G
D
. Contents
h
t
a
m 1 Introduction 1
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2 2 Preliminaries 5
v
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8 3 Rigiditytheoremsrelated toflatcones 9
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4 Riccishrinkerwithanalmostflatconeannulus 13
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.
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0 5 ProofofMaintheorems 19
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: 6 FurtherQuestions 23
v
i
X
r 1 Introduction
a
A Riccishrinker (M,g, f)isa complete Riemannian manifold (M,g)together witha smooth func-
tion f : M Rsuchthat
→ 1
Rc+Hess = g. (1.1)
f
2
BothauthorsarepartiallysupportedbyNSFgrantDMS-1510401. TheyalsoacknowledgetheinvitationtoMSRI
∗
Berkeleyinspring2016supportedbyNSFgrantDMS-1440140,wherepartofthisworkhasbeencarriedout.
1
RiccishrinkerisalsocalledasgradientshrinkingRiccisoliton. Directcalculationshowsthat (R+
∇
f 2 f) = 0. Adding f byaconstantifnecessary, weassumethroughout that
|∇ | −
R+ f 2 = f. (1.2)
|∇ |
Underthisnormalization condition, itisknown(c.f. Theorem2.2)that
e f(4π) n/2dv = eµ, (1.3)
− −
Z
M
whereµ = µ(g,1)istheentropyfunctional ofPerelman(c.f.[35]).
The Ricci shrinker (1.1) was introduced by Hamilton [22] in mid 1980’s. As critical points
of Perelman’s µ-entropy, the Ricci shrinkers play important roles in the singularity analysis of the
Ricciflow. Forexample,itisprovedbyEnders-Mu¨ller-Topping [20]thattheproper rescaling limit
ofatype-IsingularityisalwaysanontrivialRiccishrinker. Moreinformationandreferencescanbe
found in Chapter 30 of the book [18] by Chow and his coauthors. In dimension 2, the only Ricci
shrinkers are R2, S2 and RP2 with standard metrics, due to the classification of Hamilton [23]. In
dimension 3,based onthebreakthrough ofPerelman([35],[36]), through theeffortsofNaber[33],
Ni-Wallach[34]andCao-Chen-Zhu[6],etc,weknowthatR3,S2 R,S3 andtheirquotientsareall
×
thepossible Riccishrinkers.
In dimension 4 and higher, much fewer is known about Ricci shrinkers. Typically, some extra
conditions of the curvature operator(e.g. [3], [9], [7], [17], etc), or geometric properties at infin-
ity(c.f. [27], [32]) are required to draw definite geometry conclusion. We refer the readers the
surveys [4], [5]for moredetailed picture. Without such extra conditions, it isstill notclear how to
classify Riccishrinkers. However, onecan study the moduli ofRiccishrinkers. In[8], Cao-Sesum
showed the weak compactness of the moduli of Ka¨hler Ricci solitons with uniformly bounded di-
ameter and uniformly lower bound of Ricci curvature and µ-functional. The extra conditions were
gradually weakened or removed by X. Zhang [39], Weber [38], Chen-Wang [15], Z.L. Zhang [40]
andHaslhofer-Mu¨ller [24],etc.
In this article, we focus on the study of the moduli (A,H), which consists of 4d Ricci
∗
M
shrinkers which have uniform entropy lower bound and Harnack inequality of scalar curvature
on unit ball(c.f. Definition 2.3 for precise definition). Moreover, we require each Ricci shrinker
has bounded(not uniformly) scalar curvature. In light of the results of Haslhofer-Mu¨ller(c.f. The-
orem 2.6), it is known that such moduli has weak compactness. In other words, any sequence of
Riccishrinkersin (A,H)sub-converges toanorbifoldRiccishrinkerwithlocallyfinitesingular-
∗
M
ities,inthepointed-Gromov-Hausdorff topology. Nowwereversetheprocessandaskthefollowing
question:
Whatkindoforbifolds canbeapproximated byasequence ofRiccishrinkers in (A,H)?
∗
M
NotethatflatconesR4/Γarenaturallythesimplestorbifoldindimension4. Therefore,thefirststep
towardthesolutionoftheabovequestionistocheckwhethertheflatconescanbeapproximatedby
Ricci shrinkers. We completely solve this step by the following theorem, which is the main result
ofthisarticle.
Theorem 1.1 (Gap Theorem). For any A > 0 and H > 0, there exists a small positive number
ǫ = ǫ(A,H)withthefollowing property.
2
Suppose (M,p,g, f) (A,H). Thenwehave
∗
∈ M
d (M,p,g),(R4/Γ,0,g ) > ǫ (1.4)
PGH E
n o
foreveryfinitesubgroup Γ O(4)actingfreelyonS3.
⊂
NotethatTheorem1.1canbeillustrated asanǫ-regularity theorem. Namely,suppose
d (M,p,g),(R4/Γ,0,g ) < ǫ,
PGH E
n o
thenwehaveuniformcurvature,injectivityradiusestimateinsidetheunitball. Suchtypestatement
was used in the literature of studying Einstein manifolds, e.g., in Cheeger-Colding-Tian [12] and
Chen-Donaldson [14], etc. However, an essential difference here is that we do not allow rescaling
of the metric since rescaling will destroy the structure of (1.1). Theorem 1.1 was motivated by
the work of Biquard [2], Morteza-Viaclovsky [30], where they study whether an orbifold can be
approximated by Einstein metrics, with some extra assumption of the topology of the underlying
manifolds.
Asapplications ofTheorem1.1,wecanuniformly estimatethescalarcurvature RandtheRicci
potential function f.
Theorem1.2(Uniformpositive lowerboundsofscalarcurvature andpotentialfunction). For
any A > 0and H > 0,thereexistpositive constantsC ,C ,C depending on Aand H onlywiththe
a b c
followingproperties.
Suppose (M,p,g, f) (A,H). Thenthefollowinguniform estimateshold.
∗
∈ M
(a). Atbasepoint p,wehave
f C . (1.5)
a
≥
(b). Intheball B(p,1),wehave
R C . (1.6)
b
≥
(c). Onthewholemanifold M,wehave
Rf C . (1.7)
c
≥
Theorem 1.2 seems to be the first uniform positive lower bound estimates of R and f for Ricci
shrinkers. Notethattheirupperboundandnonnegativelowerboundarewellknowninliterature(c.f.
equation (1.2),inequality (2.8)andthereferencenearby).
We remark that Theorem 1.1 should be useful in the study of 4d Ricci flow singularities with
bounded positive scalar curvature. Forevery such singularity, it seems natural that all the possible
singularitymodellocateintheclosureofthemoduli (A,H),asbothrequirementsinthedefinition
M
of (A,H)aresatisfiedautomatically.
M
3
Webriefly discuss the proof of Theorem 1.1 and Theorem 1.2. The foundation of Theorem 1.1
isarigidity theorem(c.f. Theorem 3.1)for singular Eigenfunctions of thedrifted Laplace operator.
Supposevisapositivefunction satisfying
∆ v(x) = v(x), x R4 0 , (1.8)
f
∀ ∈ \{ }
where f = x2. Weshowthatvmustbecx 2 forsomeconstantc. Inotherwords,visamultipleof
|4| | |−
theGreenfunctionpoledattheorigin. ThisrigiditytheoremisintheflavoroftheclassicalBoˆcher’s
decomposition theoremforharmonicfunctions(c.f. Theorem3.9of[1]). However,therigidityhere
is even stronger, due to the ad hoc choice of f = x2 and the non-zero eigenvalue. The proof of
|4|
this rigidity theorem follows the same route as the classical Boˆcher’s theorem, by using spherical
average. Thecompleteproofisprovided insection 3.
WereducetheproofofTheorem1.1totheaforementionedrigiditytheorem. IftheRiccishrinker
(M,p,g, f)isveryclosetoaflatconeR4/Γ,thenRisclosetozerofunctiononaverylargeannulus
partB(0,δ 1) B(0,δ). WerescalethefunctionRbymultiplyingthemwithα 1,whereαisthemax-
− −
\
imumofRontheunitsphere∂B(p,1). Therescaled function isdenoted byV. Notethattheunder-
lyingmetricsarenotchangedatall. TheproofofTheorem1.1isthencarriedoutbyacontradiction
argument. SupposeTheorem1.1fails,thenwecanfindasequence(M,p,g) (A,H)converg-
i i i ∗
∈ M
ingtosome R4/Γ,0,g ,inthepointed-Gromov-Hausdorff topologyandhenceinthepointed-Cˆ -
E ∞
Cheeger-Gro(cid:16)mov topolo(cid:17)gy(c.f. Theorem 2.6 and the discussion below it). Modulo some a priori
estimates from elliptic PDE,weshowthatV isconvergent inproper topology. Moreover, thelimit
i
V is a solution of (1.8) on the smooth part of the limit flat cone and hence force can be lifted to
th∞e solution of (1.8) on R4 0 . Using the rigidity of solutions of (1.8), weobtain that V = x 2.
−
\{ } ∞ | |
However, itwillviolate our Harnack inequality assumption forthe scalar curvature. Therefore, we
obtainadesiredcontradiction toestablish theproofofTheorem1.1.
The technical difficulty of the proof of Theorem 1.1 locates in the uniform a priori estimate of
V. Actually, one has to introduce several extra auxiliary functions(c.f. Definition 2.8) for the pur-
pose of estimating V. Allofthese auxiliary functions areindicated by thesoliton identities arising
from (1.1). Therefore, it is a regularity problem of system of elliptic equations and inequalities
to obtain such estimates. These estimates are made possible due to the ad hoc structure of this
system and the important progress in the study of 4-d Ricci shrinker recently, e.g., the work of
Munteanu-Wang [31]. Onenewingredient fortheproofofTheorem1.1istoseparate therescaling
of the curvature and the rescaling of the metric. We only need to use the linear structure of the
PDEsatisfied by the curvatures. Therefore, weare able to keep the metric un-rescaled, but rescale
the curvatures as functions to obtain desired linear PDE solution V on the limit space. In the lit-
∞
erature of Ricci flow study, it seems that the rescaling of metrics and curvatures are always done
simultaneously.
We proceed to discuss the proof of Theorem 1.2. Besides Theorem 1.1, a rigidity theorem(c.f.
Theorem 3.6) of orbifold Ricci shrinkers is needed. This theorem states that every orbifold Ricci
shrinkerwithascalarcurvaturezeropointmustbeaflatcone. Itcanbeprovedbyastandardmaxi-
mumprincipleargument. BasedonthisrigiditytheoremandTheorem1.1,ourTheorem1.2follows
from a contradiction arguments. Forexample, if part (a) of Theorem 1.2 fails, then wecan extract
a sequence of Ricci shrinkers converging to an orbifold Ricci shrinker whose scalar curvature at
basepoint iszero. Consequently, weobtain asequence ofRiccishrinkers converging toaflatcone
4
R4/Γ,whichisimpossiblebyTheorem1.1. Thiscontradictionestablishes theproofofpart(a). The
remainder part of Theorem 1.2can beproved similarly, withextra difficulties which can be solved
bydelicateapplication ofmaximumprinciples onRand f.
Thispaperisorganized asfollows. Insection 2,wereviewsomeelementary resultsoftheRicci
flowandtheRiccishrinkers. Wealsointroduce important auxiliaryfunctions forthestudyofRicci
shrinkers. In section 3, we study the rigidity theorems related to flat cones. We classify all posi-
tive solutions of (1.8) which is bounded at infinity. They are nothing but the constant multiples of
Green’s function on R4 poled at the origin. Moreover, we show that any orbifold Ricci shrinker
must beflatcone ifthe scalar curvature equals zero somewhere. Insection 4, wedevelop effective
estimates forauxiliary functions onthe Riccishrinkers withan almost flatcone annulus. This sec-
tionisthetechnical coreofthispaper. Insection 5,weprovide thecomplete proofofTheorem1.1
andTheorem1.2. Finally, insection6,welistsomeopenquestions relatedtoourmaintheorems.
Acknowledgements: BothauthorsaregratefultoprofessorHaozhaoLiforinspiringdiscussion.
TheyalsothankprofessorXiuxiongChen,WeiyongHeandSongSunforhelpfulcomments. Partof
thisworkwasdonewhilebothauthorswerevisitingAMSS(AcademyofMathematicsandSystems
Science)inBeijingandUSTC(UniversityofScienceandTechnologyofChina)inHefei,duringthe
summerof2016. TheywishtothankAMSSandUSTCfortheirhospitality.
2 Preliminaries
Onacompletemanifold Mn,aRicciflowsolution isafamilyofsmoothmetricsg(t)satisfying
∂
g = 2Rc. (2.1)
∂t −
Thefollowingpseudo-locality theorem ofPerelmanisfundamental.
Theorem 2.1 (Theorem 10.3 of Perelman [35]). For every n 2 there exist δ > 0 and ǫ > 0
0
≥
dependingonlyonnwiththefollowingproperty. Let(Mn,g(t)), t [0,(ǫr )2],whereǫ (0,ǫ ]and
0 0
∈ ∈
r (0, ), be a complete solution of the Ricci flow with bounded curvature and let x M be a
0 0
∈ ∞ ∈
pointsuchthat
inj 2(x,0)+ Rm(x,0) r 2 for x B (x ,r ).
− | | ≤ 0− ∈ g(0) 0 0
Thenwehavetheinterior curvature estimate
Rm(x,t) (ǫ r ) 2 (2.2)
0 0 −
| | ≤
for x M suchthatd (x,x ) ǫ r andt (0,(ǫ r )2].
g(t) 0 0 0 0 0
∈ ≤ ∈
Notethatitisnotstatedclearlywhether M isaclosedmanifoldinPerelman’soriginaltheorem.
However,checkingtheproofcarefully,itisclearthatthestrategyoftheproofworksforRicciflows
withboundedcurvatureateachtimeslice. Rigorously,Theorem2.1inthenoncompactcasefollows
fromthecombination ofTheorem8.1ofChau-Tam-Yu[11]andTheorem3.1ofB.L.Chen[13].
5
A Ricci shrinker (M,g, f) is a complete Riemannian manifold (M,g) together with a smooth
function f : M Rsuchthat(1.1)issatisfied. Bytakingthetraceof (1.1),wehave
→
n
R+∆f = . (2.3)
2
ForaRiccishrinker (M,g, f),thereexistsasolution g(t)oftheRicciflowwithg(0) = gsuchthat
g(t) = (1 t) φt (g), (2.4)
∗
− { }
where φt isthe 1-parameter family of diffeomorphisms generated by 1 f. In particular, aRicci
1 t∇g
shrinker can be extended as an ancient Ricci flow solution. By Coroll−ary 2.5. of B.L. Chen [13],
wesee that R 0by maximum principle, even without Rm bounded condition. Moreover, by the
≥ | |
evolution equation (∂ ∆)R = 2Rc2 and thestrong maximum principle, either R > 0everywhere
t
− | |
orR 0andhenceRicci-flat. Inthelattercase,itiswellknownthat(Mn,g)isisometricto(Rn,g ),
E
≡
forexample, see Theorem 3.3ofY.Li[28]foraproof. Therefore, onanon-flat Riccishrinker, we
have
R(x)> 0, x M. (2.5)
∀ ∈
Fix Riemannian manifold (M,g), recall that Perelman’s µ(g,1)-functional is defined as the in-
fimum of W(φ,g,1) among all positive smooth functions φ with compact support on M and with
normalization condition φ2dv = 1,where
M
R
n
W(φ,g,1) , 4 φ2 +Rφ2 2φ2logφ dv n log(4π).
Z |∇ | − − − 2
Mn o
In general, for a noncompact manifold (M,g), the minimizer function of µ(g,1) may not exist.
f
However,itwasprovedbyCarrilloandNithatthefunctione−2 isalwaysaminimizerofµ(g,1),up
toadding f byaconstant.
Theorem 2.2 (Part (i) and (ii) of Theorem 1.1 of Carrillo-Ni [10]). Suppose (M,g, f) is a Ricci
shrinker. Thenwehave
µ(g,1) = W e−f+2c,g,1 = n n log(4π)+ R+ f 2 + f +c (4π)−n2e−f−cdv, (2.6)
(cid:18) (cid:19) − − 2 ZMn(cid:16) |∇ | (cid:17) o
wherecisaconstant suchthat M(4π)−n2e−f−cdv = 1.
R
By (2.6) and the Euler-Lagrangian equation satisfied by minimizer functions, we can easily
deducethatµ(g,1) = c. Therefore, wehavetheequality
(4π)−n2e−fdv = eµ(g,1). (2.7)
Z
M
In this article, we focus on the study of 4d Ricci shrinkers with uniform entropy lower bound.
Namely, we shall study the Ricci shrinker moduli (A,H), whose precise definition is stated as
M
follows.
Definition2.3. Let (A,H)bethefamilyofRiccishrinkers (M4,g, f)satisfying
M
6
1. Thescalarcurvature Rof(M,g)isbounded,
2. Forany x,y B(p,1),R(x) HR(y),
∈ ≤
3. µ(g,1) A.
≥ −
Let ∗(A,H)bethecollectionofallelementsin (A,H)excepttheGaussiansoliton R4,gE,e−|x4|2 .
M M (cid:18) (cid:19)
Byabusingofnotation, wealsosay(M,p,g, f) (A,H)if(M,g, f) (A,H)and pisamin-
∗ ∗
∈ M ∈ M
imumpointof f satisfying (2.8).
Note that in Definition 2.3, it is also required that each Ricci shrinker has bounded scalar cur-
vature. Thisisonlyfortechnicalpurposeandcouldbedroppedbyfurtherefforts(c.f. Li-Wang[29]).
WequotesomeimportantestimatesfromtheworkofR.HaslhoferandR.Mu¨ller[24],[25].
Lemma2.4 (Lemma2.1 of Haslhofer-Mu¨ller [24]). Let(Mn,g, f) be a Ricci shrinker. Then there
existsapoint p M where f attains itsinfimumand f satisfiesthequadratic growthestimate
∈
1 1 2
(d(x,p) 5n)2 f(x) d(x,p)+ √2n (2.8)
4 − + ≤ ≤ 4
(cid:16) (cid:17)
forall x M,wherea := max 0,a .
+
∈ { }
Lemma 2.5 (Lemma 2.2 of Haslhofer-Mu¨ller [24]). There exists a constant C = C(n) such that
everyRiccishrinker (Mn,g, f)with p M aminimalpointof f,
∈
VolB(p,r) Crn. (2.9)
≤
Theorem2.6(Theorem1.1ofHaslhofer-Mu¨ller [25]). Let(M,p,g, f) (A,H)beasequence
i i i i
∈ M
offourdimensional Riccishrinkers. Thenbytakingsubsequence ifnecessary, wehave
pointed Cˆ Cheeger Gromov
(M,p,g, f) − ∞− − (M ,p ,g , f ), (2.10)
i i i i
−−−−−−−−−−−−−−−−−−−−−−−−→ ∞ ∞ ∞ ∞
where(M ,p ,g , f )isanorbifoldRiccishrinker withlocallyfinitesingular points.
∞ ∞ ∞ ∞
Notethattheconvergence topology in(2.10)wasstatedas“pointed-orbifold-Cheeger-Gromov”
topology. Let us say a few more words for its precise meaning. In fact, (2.10) first means that
(M,p,d)converges toalength space (M ,p ,d ), whered isthe distance structure induced by
i i i i
∞ ∞ ∞
g. Thenonecandecompose thelimitspace M intoregular part (M )andsingular part (M ).
i
∞ R ∞ S ∞
Hereregular part (M )isasmooth manifold equipped withasmooth metricg . Locally around
R ∞ ∞
each regular point, the metric structure determined by g is identical to d . The singular part is a
∞ ∞
collectionofdiscretepoints. Theregularpart (M )hasanexhaustion K bycompactsetsK .
R ∞ ∪∞j=1 j j
For each compact set K = K for some j, one can find diffeomorphisms ϕ from K to ϕ (K), a
j K,i K,i
subsetof M suchthat
i
d(ϕ (x),p) d (x,p ), x K;
i K,i i
→ ∞ ∞ ∀ ∈
C
ϕ (g) ∞ g , on K;
∗K,i i −→ ∞
ϕ (f) C∞ f , on K.
∗K,i i −→ ∞
7
Although in general the global distance structure induced by g may not be the same as d , this
∞ ∞
difference does not happen whenever M is an orbifold with isolated singularities since (M ) is
∞ R ∞
geodesic convex.
Recall that M is an orbifold with discrete singularities. For each singular point, i.e., a point
∞
p (M ), one can find a small δ = δ(p), an open neighborhood U of p, and a smooth nonde-
∈ S ∞
generate map π : B(0,δ) 0 U p such that h = π (g ) is a smooth metric on B(0,δ) 0
∗
\{ } → \{ } ∞ ∞ \{ }
andlim h (x)exits. Moreover,bysettingh (0) = lim h (x),thenh isasmoothmetricon
x 0 x 0
→ ∞ ∞ → ∞ ∞
B(0,δ). Notethath isΓ-invariant,whereΓisthelocalorbifoldgroupof p. Thetriple(B(0,δ),π,U)
∞
iscalledanorbifoldchartaround p,h iscalledtheorbifoldliftingofthemetrictensorg .
∞ ∞
By an orbifold Ricci shrinker (M ,g , f ) we mean that the identity Rc + Hessf = 1g
2
∞ ∞ ∞ ∞ ∞ ∞
holdssmoothlyonanyorbifoldchartafterthelifting, where f isasmoothfunctionintheorbifold
∞
sense. In other words, f isa smooth function in each orbifold chart. Clearly, the scalar curvature
∞
R is also a smooth function in the orbifold sense. By abuse of notation, we use R (p) to denote
∞ ∞
thevalueπ (R )(0).
∗
∞
Thefollowingbeautiful workofO.Munteanu andJ.P.Wangisalsoimportantforus.
Theorem2.7. (Theorem2.5and2.6ofMunteanu-Wang[31])Suppose(M,g, f)isa4-dimensional
Riccishrinkerwithboundedscalarcurvature. Thenthereisaconstant Ldependingon M suchthat
Rm + Rm
sup | | |∇ | < L. (2.11)
R
M
In particular, (2.11) implies that each soliton in (A,H) has bounded curvature and Theo-
∗
M
rem2.1canbeapplied.
Now for a general shrinking soliton (M,g, f), as it can be regarded as a normalized Ricci flow
solution, wehavethefollowingellipticequations where∆ = ∆ , f .
f
−h∇ ∇ i
n
∆ f = f, (2.12)
f
2 −
n
∆ f 1 = f 1 2Rf 3+ 2 f 2. (2.13)
f − − − −
− (cid:18) − 2(cid:19)
Theparticular caseof(2.13)indimensionfouris
∆ f 1 = f 1 2Rf 3. (2.14)
f − − −
−
Thefollowingevolution equations arewell-known(c.f. forexample,Munteanu-Wang [31]):
∆ R= R 2Rc2, (2.15)
f
− | |
Rc2 Rc2 Rc4 RRc RcR2 Rm(Rc,Rc)
∆ | | = | | +2| | +2|∇ −∇ | 4 . (2.16)
f
R R R2 R3 − R
∆ R = R 2R R , (2.17)
f ij ij ikjl kl
−
1
R = R f = R, (2.18)
k jk jk k j
∇ 2∇
8
Definition2.8. LetαbethemaximumofRontheunitsphere∂B(p,1). Wedefineauxiliaryfunctions
asfollows:
Rc2 R Rc R R Rc2
U , | | , V , , Z , | ∇ − ∇ | . (2.19)
R α R3
Lemma2.9. Theauxiliary functions satisfythefollowingellipticrelationships.
∆ V = V 2UV, (2.20)
f
−
∆ U (1 4Rm)U +2 Z+U2 . (2.21)
f
≥ − | |
(cid:16) (cid:17)
Proof. Theequation (2.20)followsfrom(2.15). Theinequality (2.21)followsfrom(2.16). (cid:3)
3 Rigidity theorems related to flat cones
Weinvestigate thepositivesolutionoftheequation
∆ v= v (3.1)
f
on R4/Γ 0 , where ∆ = ∆ f, and f = r2 = x2. By lifting to the orbifold covering,
{ }\{ } f − h∇ ∇·i 4 |4|
it suffices to study the solution of (3.1) for the special case Γ = 1 , where R4/Γ 0 becomes
{ } { }\{ }
punctured Euclidean space. Allthe results in this section hold forthe general dimension n, but we
will only focus on dimension 4 for not distracting the readers’ attention from the main stream of
thispaper. Forsimplicityofnotation,weuseB(r)todenotetheballofradiusrinR4centeredatthe
origin0.
Theorem 3.1 (Rigidity of eigenfunctions). Suppose v > 0 satisfies (3.1) on R4/Γ 0 and v is
{ }\{ }
bounded atinfinity, i.e.,limsupv(x) < . Thenwehave
∞
x
→∞
v = br 2 (3.2)
−
forsomeconstant b.
Wefirstconsider the possible radial solutions. Letv(x) = h(r) be anradial solution of ∆ v = v.
f
Thenfromdirectcomputation, wehave
3 r
h (r)+ h (r) = h(r) (3.3)
′′ ′
r − 2!
for r > 0. (3.3) is a second order linear ODE,the basis consists of er2/4r 2 and r 2. Therefore the
− −
generalradialsolution of (3.1)isc er2/4r 2+c r 2 forsomeconstants c ,c .
1 − 2 − 1 2
9
Nowforanysolutionvof (3.1),wedefineitsspherical average
1
A[v](r) = vdσ
Ω3r3 ZSr
where Ω is the volume of unit S3. We have the following decomposition lemmas, whose proofs
3
aresimilartothoseinTheorem3.9ofAxler-Bourdon-Ramey [1].
Lemma3.2. If∆ v= v,then∆ A[v] = A[v].
f f
Proof. Fromthechangeofvariable, A[v](r) = 1 v(rw)dw,wheredwisthevolumeformonthe
Ω3 S3
unitsphereS3. Therefore, R
1 d2 3 r d
∆ A[v]= + v(rw)dw
f
Ω3 ZS3 dr2 r − 2! dr!
1 d2 3 r d ∆
= + + S3 v(rw)dw
Ω3 ZS3 dr2 r − 2! dr r2 !
1 1
= ∆ v(rw)dw = v(rw)dw = A[v],
f
Ω3 ZS3 Ω3 ZS3
where the second identity is true since ∆ v(rw)dw = 0 and the third identity holds since we
S3 S3
have∆ = d2 + 3 d + ∆S3 and v, f =R r dv. (cid:3)
dr2 r dr r2 h∇ ∇ i 2dr
Lemma3.3. There exists aconstant c > 0such that for every positive solution v of (3.1) on B(1),
wehave
v(x) > cv(y)
forany0 < x = y 1/2.
| | | | ≤
Proof. When x = y = 1/2, the conclusion follows from the standard Harnack inequality for the
| | | |
ellipticoperator ∆ Id,see[21,Theorem8.20]. For x = y = a 1/2,wesetv˜(x) = v(ax),then
f
− | | | | ≤
∆ v˜ = a2v˜
f˜
where f˜(x) = f(ax). Again, we have the Harnack inequality for v˜ whenever x = y = 1/2. The
| | | |
Harnackconstantisindependentofaasthecoefficientsoftheaboveellipticequationsareuniformly
controlled. (cid:3)
Lemma3.4. Ifvisapositivesolutionof (3.1)onB(1) 0 suchthatvtendstoaconstantas x 1,
\{ } | | →
thenv = A[v].
Proof. Fromlemma3.3,thereexistsaconstantc (0,1)suchthatv cA[v] > 0on B(1/2). Onthe
∈ −
other hand, since v cA[v] 0 as x 1, by the strong maximal principle on B(1) B(1/2), we
− → | | → \
conclude that v cA[v] > 0 on B(1). Similarly, v cA[v] > cA[v cA[v]] = cA[v] c2A[v] since
− − − −
v cA[v]satisfiesthesamecondition asv. Byiteration, weconclude that
−
v > g(m)(c)A[v]
foranyintegerm > 0,whereg(t) = c+t(1 c),g(m) isthem-thiteration ofg. Nowasg(m)(c) 1
− →
ifm ,wehavev A[v]. Ontheother hand, since A[v A[v]] = 0,weconclude thatv = A[v]
→ ∞ ≥ −
on B(1). (cid:3)
10