Ends, fundamental tones, and capacities of minimal submanifolds via extrinsic comparison theory

0.9 MB·

by  Vicent Gimeno

ENDS, FUNDAMENTAL TONES, AND CAPACITIES OF MINIMAL
SUBMANIFOLDS VIA EXTRINSIC COMPARISON THEORY
VICENTGIMENOANDS.MARKVORSEN
ABSTRACT. Westudythevolumeofextrinsicballsandthecapacityofextrinsicannuli
in minimal submanifolds which are properly immersed with controlled radial sectional
curvaturesintoanambientmanifoldwithapole. Thekeyresultsareconcernedwiththe
comparisonofthosevolumesandcapacitieswiththecorrespondingentitiesinarotation-
4
allysymmetricmodelmanifold. Usingtheasymptoticbehaviorofthevolumesandca-
1
pacitieswethenobtainupperboundsforthenumberofendsaswellasestimatesforthe
0
fundamentaltoneofthesubmanifoldsinquestion.
2

n
a
J
1. INTRODUCTION
7
Let M be a completenon-compact Riemannian manifold. Let K ⊂ M be a compact

] set with non-empty interior and smooth boundary. We denote by E (M) the number of
G K
connected components E ,··· ,E of M \K with non-compact closure. Then M
D 1 EK(M)
hasE (M)ends{E }EK(M)withrespecttoK(seee.g.[GSC09]),andtheglobalnumber
. K i i=1
h ofendsE(M)isgivenby
t
a (1.1) E(M)= sup E (M) ,
K
m K⊂M
[ whereK rangesonthecompactsetsofM withnon-emptyinteriorandsmoothboundary.

The number of ends of a manifold can be bounded by geometric restrictions. For ex-
1
ample, in the particular setting of an m−dimensional minimal submanifold P which is
v
9 properlyimmersedintoEuclideanspaceRn,thenumberofendsE(P)isknowntobere-
2 latedtotheextrinsicpropertiesoftheimmersion.Indeed,V.G.Tkachevprovedin[Tka94,
3 Theorem 2] (see also [Che95]) that for any properly immersed m−dimensional minimal
1 submanifold P in Rn with finite volume growth V (P) < ∞ the number of ends is
. w0
1 boundedfromaboveby
0
4 (1.2) E(P)≤CmVw0(P) ,
1 whereC =1(C =2mintheoriginal[Tka94])andthevolumegrowthV (P)is
: m m w0
v Vol(cid:0)P ∩BRn(o)(cid:1)
i (1.3) V (P)= lim R .
X w0 R→∞ Vol(cid:0)BRm(o)(cid:1)
R
r
a HereVol(cid:0)BRm(o)(cid:1)isthevolumeofageodesicballBRm(o)ofradiusRcenteredatoin
R R
Rm.Theinequality(1.2)thusshowsasignificantrelationbetweenthenumberofends(i.e.
atopologicalproperty)andthebehaviorofaquotientofvolumes(i.e. ametricproperty).
MotivatedbyTkachev’sapplicationofthevolumequotientappearinginequation(1.3),
wewillconsiderthecorrespondingfluxquotientandcapacityquotientoftheminimalsub-
manifolds. These quotients are constructed in the same way as indicated by the volume
quotient but here we generalize the setting as well as Tkachev’s result to minimal sub-
manifolds in more general ambient spaces as alluded to in the abstract. Specifically we
assumethattheminimalimmersiongoesintoanambientmanifoldN withapoleandwith
2010MathematicsSubjectClassification. Primary53A,53C.
Keywordsandphrases. FirstDirichleteigenvalue,Capacity,Effectiveresistance,Minimalsubmanifolds.
WorkpartiallysupportedbyDGIgrantMTM2010-21206-C02-02.
1
2 V.GIMENOANDS.MARKVORSEN
sectionalcurvaturesK boundedfromabovebytheradialcurvaturesK ofarotationally
N w
symmetricmodelspaceMn =R+×Sn−1,withwarpedmetrictensorg constructedus-
w 1 Mwn
ingapositivewarpingfunctionw :R+ →R+insuchawaythatg =dr2+w(r)2g
Mn Sn−1
w 1
isalsobalancedfrombelow(see[MP06]and§3foraprecisedefinitions).
Ourgeneralizationofinequality(1.2)isthencethefollowing:
Theorem 1.1. Let ϕ : Pm → Nn be a proper minimal and complete immersion into a
n−dimensional ambient manifold Nn which possesses a pole o ∈ Nn and its sectional
curvaturesK atanypointp ∈ N areboundedbyabovebytheradialcurvaturesK of
N w
abalancedfrombelowmodelspaceMn
w
w(cid:48)(cid:48)
(1.4) K (p)≤K (r(p))=− (r(p)) .
N Mwn w
Supposemoreover,thatw(cid:48) >0andthereexistR suchthatK (R)≤0foranyR>R .
0 Mn 0
w
Then,thenumberofendsE (P)withrespecttotheextrinsicballD =P ∩BN(o)for
DR R R
R>R isboundedfromaboveby
0
(cid:32) 2 (cid:33)m(cid:32)(cid:82)tw(s)m−1ds(cid:33) Vol(D )
(1.5) E (P)≤ 0 t ,
DR 1− R tm/m Vol(Bw)
t t
foranyt>R.
Usingtheabovetheoremwecanestimatetheglobalnumberofendsasfollows:
Corollary1.2. Undertheassumptionsoftheorem1.1,supposemoreover
(cid:32)(cid:82)tw(s)m−1ds(cid:33)
(1.6) limsup 0 =C <∞ ,
tm/m w
t→∞
andsupposealsothatthesubmanifoldhasfinitevolumegrowth,namely
Vol(D )
(1.7) Vol (P)= lim t <∞ .
w t→∞Vol(Bw)
t
Then
(1.8) E(P)≤2mC Vol (P) .
w w
Remarka. Ifwechoosew(t) = w (t) = t,themodelspacebecomesRm,whichisbal-
0
ancedfrombelow,andthehypothesisoftheorem1.1arethereforeautomaticallyfulfilled
foranycompleteminimalsubmanifoldproperlyimmersedinaCartan-Hadamardambient
manifold. Inequality(1.5)becomes
(cid:32) (cid:33)m
2 Vol(D )
(1.9) E (P)≤ t ,
DR 1− R V tm
t m
ForanyR>0andanyt>R,beingV thevolumeofageodesicballofradius1inRm.
m
Frominequality(1.6)weget
(1.10) C =1 .
w0
Thusinequality(1.8)becomes
Vol(D )
(1.11) E(P)≤2m lim t ,
t→∞ Vmtm
whichistheoriginalinequalityobtainedbyTkachev(inequality(1.2)),butnowinequality
(1.11) is valid for any minimal submanifold properly immersed in a Cartan-Hadamard
ambientmanifoldwithfinitevolumegrowth.
ENDS,FUNDAMENTALTONES,ANDCAPACITIES 3
FIGURE 1. TwoexamplesofextrinsicannuliinR3: Acatenoidonthe
left and the singly periodic Scherk surface on the right. The extrinsic
annuliareconstructedbycuttingthesurfaceswithtwospheres(withthe
same center but of different radii) in the ambient manifold (R3). The
catenoidhastwoendsandfinitetotalcurvature. Hence,bytheorem1.4,
the capacity of the extrinsic annulus of the catenoid is greater than the
capacity of the corresponding annulus of the Euclidean 2-plane but is
smaller than two times that capacity. The same is true for the extrinsic
annulusofthesinglyperiodicScherksurface(wereferthereadertothe
introductionof[MW07]fortheareagrowthofthesinglyperiodicScherk
surface).
In[GP12,Che95]arealsoobtainedlowerboundsforthenumberofends,butwenote
that those lower bounds seem to need stronger assumptions: Dimension greater than 2,
orembeddednessoftheendsandcodimension1,decayonthesecondfundamentalform,
andarotationallysymmetricambientmanifold. Asacounterpart,thoselowerboundsare
associatedtotheso-calledgaptypetheorems.
Combining the results of [GP12, Theorem 3.5] and corollary 1.2, and taking into ac-
counttheroleofsectionalcurvaturesofthemodelspace(see[GP12,Proposition2.6])we
have
Corollary1.3. Letϕ : Pm → Mn beaminimalandproperimmersionintoabalanced
w
frombelowmodelspaceMn withincreasingwarpingfunctionw satisfyingthefollowing
w
conditions:
(cid:32)(cid:82)tw(s)m−1ds(cid:33)
limsup 0 =C <∞ ,
tm/m w
t→∞
thereexistR suchthatforanyR>R
(1.12) 0 0

−w(cid:48)(cid:48)(R) ≤0
w(R)
1−(w(cid:48)(R))2
 ≤0
w(R)2
Suppose moreover that m > 2, that the center o of Mn satisfies ϕ−1(o ) (cid:54)= ∅ and
w w w
thatthenormofthesecondfundamentalform(cid:107)BP(cid:107)oftheimmersionisboundedforlarge
rby
(cid:15)(r)
(1.13) (cid:107)BP(cid:107)≤ ,
w(cid:48)(r)w(r)
where(cid:15)isapositivefunctionsuchthat(cid:15)(r)→0whenr →∞.
Thenthenumberofendsisboundedfrombelowandfromaboveby
(1.14) Vol (P)≤E(P)≤2mC Vol (P) .
w w w
Byusingourresultsaboutthebehaviorofthecomparisonquotientswecanalsoestimate
thecapacityofanextrinsicannulusA = P ∩(cid:0)BN(o)\BN(o)(cid:1)(seefigure1and,§2
ρ,R R ρ
and§3foraprecisedefinitionofcapacityandextrinsicannulus):
4 V.GIMENOANDS.MARKVORSEN
Theorem1.4. Letϕ : Pm → Rn denoteacompleteandproperminimalimmersioninto
theEuclideanspaceRn. Then,foranyR > ρ > 0,thecapacityoftheextrinsicannulus
A isboundedby
ρ,R
Vol(D ) Cap(A ) Vol(D )
(1.15) ρ ≤ ρ,R ≤ R ,
V ρm Cap(ARm) V Rm
m ρ,R m
where Cap(ARm) is the capacity of the geodesic annulus ARm in Rm of inner radius ρ
ρ,R ρ,R
andouterradiusR.
Remarkb. Since,fromTheorem2.1,thequotient Vol(Ds) isanon-decreasingfunctionon
Vmsm
s,wecanstateTheorem1.4inthelimitcase(ρ → 0andR → ∞)andinequality(1.15)
therebecomes
Cap(A ) Vol(D )
(1.16) 1≤ ρ,R ≤ lim R =V (P) .
Cap(ARρ,mR) R→∞ VmRm w0
When we deal with a minimal surface Σ ⊂ R3 which is properly embedded into the
EuclideanspaceR3thelimit
Vol(Σ∩BR3(o))
(1.17) V (Σ)= lim R
w0 R→∞ πR2
iswellunderstood. Forinstance,theabovelimitcorrespondstothenumberofendsifthe
surfaceΣhasfinitetotalcurvature.
Remarkc. Inordertoboundthecapacityquotient,ourtheoremsdonotmakeuseofthe
volume quotient as in Theorem 1.4, but instead they make use of the flux quotient (see
Theorems 2.2 and 2.3). In the special case when the ambient manifold is Rn (such as
inTheorem1.4)thevolumequotientagreeshoweverwiththefluxquotient(seeequation
(6.28)andtheorem6.1).
1.1. Outline of the paper. In §2 we show our main theorems concerning the flux quo-
tients, the volume quotients, and the capacity quotients. In §3 we state the preliminary
concepts in order to prove the main theorems of §2 in §4. This allows us then to prove
Theorem 1.1 and Corollary 1.2 in §5. Finally, in §6, we present several corollaries and
examplesofapplicationsoftheextrinsictheoryandresultswhichhavebeenestablishedin
§2.
2. EXTRINSICTHEORY: FLUX,CAPACITYANDVOLUMECOMPARISONFOR
EXTRINSICBALLS
Let (Mn,g) be a Riemannian manifold. For any oriented hypersurface Σ ⊂ M with
unitnormalvectorfieldν,wedefinethefluxF (Σ)ofthevectorfieldX throughΣby
X
(cid:90)
(2.1) F (Σ):= (cid:104)X,ν(cid:105)dµ ,
X Σ
Σ
wheredµ istheassociatedRiemanniandensitydeterminedbythemetricg =i∗g(being
Σ Σ
i:Σ→M theinclusionmap).
By the divergence theorem (see [Cha93] for instance), if one has an oriented domain
Ω in M with smooth boundary ∂Ω, and the vector field X is C1 in Ω and with compact
supportinΩ,thefluxofX through∂ΩisrelatedtothedivergenceofX by
(cid:90) (cid:90)
(2.2) divXdµ= (cid:104)X,ν(cid:105)dµ =F (∂Ω) .
∂Ω X
Ω ∂Ω
Given a smooth function u : M → R, we can also define the flux of a function u,
but then the flux J (t) is the flux of the gradient ∇u (i.e. the metric dual vector to du,
u
du(X)=(cid:104)∇u,X(cid:105))throughthelevelsetΣu :={x∈M|u(x)=t}sothat:
t
ENDS,FUNDAMENTALTONES,ANDCAPACITIES 5
(2.3) J (t):=F (Σu) .
u ∇u t
Takingintoaccountthattheunitnormalvectorfieldν ofΣu isν = ∇u ,itiseasyto
t |∇u|
seethat
(cid:90)
(2.4) Ju(t)= |∇u|dµΣu .
t
Σt
Observe moreover, that by the Sard theorem and by the regular set theorem we need
nofurtherrestrictionsonthesmoothnessofΣu andonthesmoothnessoftheunitnormal
t
vectorfieldν.
The overall goal of this work is to characterize the isoperimetric inequalities for ex-
trinsic balls, and the capacity of minimal submanifolds in terms of the flux of extrinsic
distancefunctions. Actuallyweareinterestedonthefluxoftheextrinsicdistancefunction
on minimal submanifolds in an ambient manifold N which possesses a pole and has the
radial curvatures bounded form above by the radial curvatures of rotationally symmetric
modelspaceK ≤ K = −w(cid:48)(cid:48),see[MP06]orsection3ofthispaperforprecisedef-
N Mwn w
initions. Itisthebehaviorofthisparticularfluxthatallowsustostudythemeanexittime
function,thecapacity,theconformaltype,thefundamentaltone,andinspecialcasesalso
thenumberofendsofthesubmanifold.
2.1. Fluxandvolumecomparison: isoperimetricinequalitiesandthemeanexittime
function. Givenanisometricimmersionϕ:P →(N,o)intoamanifoldwithapoleo∈
N, thefluxJ oftheextrinsicdistancefunctionr (i.e., therestrictionbytheimmersion
r o
oftheambientdistancefunctiontothesubmanifold)isgivenby
(cid:90)
J (R)= |∇Pr |dµ ,
r o
∂DR
where ∂D is the level set ∂D = r−1(R), and therefore, D = r−1([0,R)) is the
R R o R o
extrinsicballofradiusR.
Whentheimmersionisminimalandtheambientmanifoldhasitsradialsectionalcurva-
turesK boundedfromabovebytheradialsectionalcurvaturesofarotationallysymmet-
N
ricmodelspaceMnthatisbalancedfrombelow(see[MP06]andsection3),K ≤K ,
w N Mwn
we can compare the volume quotient Vol(DR) and the flux quotient Jr(R) . The volume
Vol(Bw) Jw(R)
quotient is the quotient between the volumRe of a extrinsic ball D rof radius R in Pm
R
andthevolumeofageodesicballBw ofthesameradiusRinMm. Thefluxquotientis
R w
thequotientbetweenthefluxoftheextrinsicdistanceinPm andthefluxofthegeodesic
distanceinMm. Thesetwoquotientsarerelatedbythefollowingtheorem
w
Theorem2.1. Letϕ : Pm −→ Nn beanisometric,proper,andminimalimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nn withapoleo ∈ N . Letussupposethattheo−radialsectionalcurvaturesofN are
boundedfromaboveby
w(cid:48)(cid:48)(r)
K (σ )≤− (ϕ(x)) ∀x∈P ,
o,N x w(r)
andthemodelspaceMmisbalancedfrombellow. Then
w
(1) J (R)isrelatedwithVol(D )by
r R
Vol(D ) J (R)
(2.5) R ≤ r .
Vol(Bw) Jw(R)
R r
(2) Thefunctions Vol(DR) and Jr(R) arenondecreasingfunctionsonR.
Vol(Bw) Jw(R)
R r
6 V.GIMENOANDS.MARKVORSEN
(3) DenotingbyEP(x)themeantimeforthefirstexitfromtheextrinsicballD (o)
R R
foraBrownnianparticlestartingato ∈ Pm,anddenotingbyEw themeanexit
R
timefunctionfortheR−ballBwinthemodelspaceMm,ifequalityholdsin(2.5)
R w
forsomefixedradiusR > 0, thenforanyx ∈ D , EP(x) = Ew(r(x)), where
R R R
r(x)theextrinsicdistancefromotothepointx∈P.
2.2. Capacityandfluxcomparison: conformaltype. GivenacompactsetF ⊂M ina
RiemannianmanifoldM andanopensetG⊂M containingF,wecallthecouple(F,G)
acapacitor. Eachcapacitorthenhasitscapacitydefinedby
(cid:90)
(2.6) Cap(F,G):=inf (cid:107)∇u(cid:107)dµ ,
u
G\F
wheretheinf istakenoverallLipschitzfunctionsuwithcompactsupportinGsuchthat
u=1onF.
When G is precompact, the infimum is attained for the function u = Ψ which is the
solutionofthefollowingDirichletprobleminG−F:

∆Ψ=0

(2.7) Ψ| =0
∂F
Ψ| =1
∂G
Fromaphysicalpointofview,thecapacityofthecapacitor(F,G)representsthetotal
electriccharge(generatedbytheelectrostaticpotentialΨ)flowingintothedomainG−F
throughtheinteriorboundary∂F. Sincethetotalcurrentstemsfromapotentialdifference
of 1 between ∂F and ∂G, we get from Ohm’s Law that the effective resistance of the
domainG−F is
1
(2.8) R (G−F)= .
eff Cap(F,G)
Theexactvalueofthecapacityofasetisknownonlyinafewcases,andsoitsestima-
tioningeometricaltermsisofgreatinterest,notonlyinelectrostatic,butinmanyphysical
descriptionsofflows,fluids,heat,orgenerallywheretheLaplaceoperatorplaysakeyrole,
see[CFG05,HPR12].
Givenacapacitor(F,G),ifwehaveasmoothfunctionuwithu=aon∂F andu=b
on∂G. Thecapacityandthefluxarethenrelatedby(see[Gri99b]):
(cid:32) (cid:33)−1
(cid:90) b ds
(2.9) Cap(F,G)≤ .
J (s)
a u
Inthispaperweareinterestedontheo-centeredextrinsicannulusA (o) ⊂ Pm for
ρ,R
0<ρ<Rgivenby
(2.10) A (o):=D (o)−D (o) .
ρ,R R ρ
Tobemoreprecise,weareinterestedonthebehaviorofthefluxandthecapacityofthose
extrinsicdomains. Inthefollowingtheoremsweprovideupperandlowerboundsforthe
capacityquotientintermsofthefluxquotient.
Theorem2.2. Letϕ : Pm −→ Nn beanisometric,proper,andminimalimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nn with a pole o ∈ N and satisfying ϕ−1(o) (cid:54)= ∅. Let us suppose that the o−radial
sectionalcurvaturesofN areboundedfromaboveby
w(cid:48)(cid:48)(r)
K (σ )≤− (ϕ(x)) ∀x∈P ,
o,N x w(r)
ENDS,FUNDAMENTALTONES,ANDCAPACITIES 7
andthewarpingfunctionwsatisfies
w(cid:48) ≥0 .
Then
J (ρ) Cap(A )
(2.11) r ≤ ρ,R ,
Jw(ρ) Cap(Aw )
r ρ,R
whereAw istheintrinsicannulusinMm.
ρ,R w
Theorem2.3. Letϕ : Pm −→ Nn beanisometric,proper,andminimalimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nn withapoleo ∈ N . Letussupposethattheo−radialsectionalcurvaturesofN are
boundedfromaboveby
w(cid:48)(cid:48)(r)
K (σ )≤− (ϕ(x)) ∀x∈P ,
o,N x w(r)
andthemodelspaceMmisbalancedfrombellow. Then
w
Cap(A ) J (R)
(2.12) ρ,R ≤ r ,
Cap(Aw ) Jw(R)
ρ,R r
whereAw istheintrinsicannulusinMm. Moreover,ifequalityholdsin(2.12)forsome
ρ,R w
fixedR>0,thenD isaminimalconeinNn.
R
Geometric estimates of the capacity are sufficient to obtain large scale consequences
suchasastheparabolicorhyperboliccharacterofthemanifold, [Ich82b,Ich82a,MP03,
MP05]. Wenoteherethefollowingimportantequivalentconditionsabouttheconformal
type:
Theorem A. Let (M,g) be a given Riemannian manifold, Then the following conditions
areequivalent
• ThereisaprecompactopendomainK inM,suchthattheBrownianmotionX
t
startingfromK doesnotreturntoK withprobability1,i.e.
(2.13) P {ω|X (ω)∈K forsomet>0}<1
x t
• M haspositivecapacity: ThereexistinM acompactdomainK,suchthat
(2.14) Cap(K,M)>0
• M has finite resistance to infinity: There exist in M a compact domain K, such
that
(2.15) R (M −K)<∞
eff
Amanifoldsatisfyingtheconditionsoftheabovetheoremwillbecalledahyperbolic
manifold,otherwiseitiscalledaparabolicmanifold.
Asaconsequenceoftheabovetheoremwecanstatethefollowingcorollaryforminimal
submanifolds:
Corollary2.4. Letϕ:Pm −→Nn beanisometric,proper,andminimalimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nn withapoleo ∈ N . Letussupposethattheo−radialsectionalcurvaturesofN are
boundedfromaboveby
w(cid:48)(cid:48)(r)
K (σ )≤− (ϕ(x)) ∀x∈P ,
o,N x w(r)
andthewarpingfunctionwsatisfies
w(cid:48) ≥0 .
Then
8 V.GIMENOANDS.MARKVORSEN
(1) IfMmisahyperbolicmanifold,thenP isahyperbolicmanifold.
w
(2) Inconsequence,ifP isparabolic,thenMmisalsoparabolic.
w
Since Jr(R) and Vol(DR) are non-decreasing functions under our hypothesis, we can
Jw(R) Vol(Bw)
r R
define two expressions which are analogous to the projective volume defined by V. G.
Tkachevin[Tka94]
Definition2.5. Givenϕ:Pm →NnanimmersionintoamanifoldN withapoleo∈N.
Thew-fluxFlux (P)andthew-volumeVol (P)ofthesubmanifoldP aredefinedby:
w w
J (R)
Flux (P):= sup r ,
w Jw(R)
R∈R+ r
(2.16)
Vol(D )
Vol (P):= sup R .
w Vol(Bw)
R∈R+ R
WewillsaythatP hasfinitew−flux(resp. finitew−volume)ifandonlyifFlux (P)<
w
∞(orVol (P)<∞).
w
We refer to theorem 6.1 for the relation between the w−flux and the w−volume of a
submanifold.
FromtheoremAandtheorem2.3wecannowstatethatforminimalsubmanifoldswith
finitew−fluxwehave:
Corollary2.6. Letϕ : Pm −→ Nn beanisometric,properandminimalimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nn withapoleo∈Nn. Letussupposethattheo−radialsectionalcurvaturesofNn are
boundedfromaboveasfollows
w(cid:48)(cid:48)(r)
K (σ )≤− (ϕ(x)) ∀x∈P ,
o,N x w(r)
andthatthemodelspaceMmisbalancedfrombelow. SupposemoreoverthatP hasfinite
w
w−flux. Then
(1) IfMmisaparabolicmanifold,thenP isaparabolicmanifold.
w
(2) IfP isanhyperbolicmanifold,thenMnisanhyperbolicmanifold.
w
Joiningtheprevioustwocorollariestogetherweget:
Corollary2.7. Letϕ : Pm −→ Nn beanisometric,properandminimalimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nnwithapoleo∈Nn. Letussupposethattheo−radialsectionalcurvaturesofNnare
boundedfromabove,
w(cid:48)(cid:48)(r)
K (σ )≤− (ϕ(x)) ∀x∈P ,
o,N x w(r)
thatthewarpingfunctionwsatisfies
w(cid:48) ≥0 ,
andthatthemodelspaceMmisbalancedfrombelow,andthatP hasfinitew−flux. Then
w
P ishyperbolic(parabolic)ifandonlyifMmishyperbolic(parabolic).
w
3. PRELIMINAIRES
Weassumethroughoutthepaperthatϕ : Pm −→ Nn isanisometricimmersionofa
completenon-compactRiemannianm-manifoldPmintoacompleteRiemannianmanifold
Nnwithapoleo∈Nn. Recallthatapoleisapointosuchthattheexponentialmap
exp : T Nn →Nn
o o
isadiffeomorphism.
ENDS,FUNDAMENTALTONES,ANDCAPACITIES 9
For every x ∈ Nn −{o} we define r(x) = r (x) = dist (o,x), since o is a pole
o N
thisdistanceisrealizedbythelengthofauniquegeodesicfromotox,whichistheradial
geodesicfromo. Wealsodenotebyr| orbyr thecompositionr◦ϕ : P → R ∪{0}.
P +
ThiscompositioniscalledtheextrinsicdistancefunctionfromoinPm.
WiththeextrinsicdistancewecanconstructtheextrinsicballD (o)ofradiusRcen-
R
teredatoas
D (o):={x∈P :r(ϕ(x))<R} .
R
Since∂D (o)=Σr,thefluxoftheextrinsicdistancefunctionronP is
t t
(cid:90)
J (t)= |∇Pr|dρ ,
r
∂Dt
wherethegradientsofrinN andr| inP aredenotedby∇Nrand∇Pr,respectively.
P
Thesetwogradientshavethefollowingbasicrelation,byvirtueoftheidentification,given
anypointx∈P,betweenthetangentvectorfieldsX ∈T P andϕ (X)∈T N
x ∗x ϕ(x)
(3.1) ∇Nr =∇Pr+(∇Nr)⊥,
where(∇Nr)⊥(ϕ(x))=∇⊥r(ϕ(x))isperpendiculartoT P forallx∈P.
x
Wenowpresentthecurvaturerestrictionswhichconstitutethegeometricframeworkof
thepresentstudy.
Definition3.1. LetobeapointinaRiemannianmanifoldN andletx ∈ N −{o}. The
sectionalcurvatureK (σ )ofthetwo-planeσ ∈T N isthencalledao-radialsectional
N x x x
curvatureofN atxifσ containsthetangentvectortoaminimalgeodesicfromotox.
x
WedenotethesecurvaturesbyK (σ ).
o,N x
3.1. Model spaces. Throughout this paper we shall assume that the ambient manifold
Nn has its o-radial sectional curvatures K (x) bounded from above by the expression
o,N
K (r(x)) = −w(cid:48)(cid:48)(r(x))/w(r(x)), whicharepreciselytheradialsectionalcurvaturesof
w
thew-modelspace Mm wearegoingtodefine.
w
Definition3.2(See[O’N83,Gri99a,GW79]). Aw−modelMmisasmoothwarpedprod-
w
uctwithbaseB1 = [0,Λ[⊂ R(where0 < Λ ≤ ∞), fiberFm−1 = Sm−1 (i.e. theunit
1
(m−1)-spherewithstandardmetric),andwarpingfunctionw: [0,Λ[→ R ∪{0},with
+
w(0) = 0, w(cid:48)(0) = 1, and w(r) > 0 for all r > 0. The point o = π−1(0), where π
w
denotestheprojectionontoB1,iscalledthecenterpointofthemodelspace. IfΛ = ∞,
theno isapoleofMm.
w w
Proposition 3.3. The simply connected space forms Km(b) of constant curvature b are
w−modelswithwarpingfunctions
√
√1 sin( br) ifb>0
 b
w (r)= r ifb=0
b
√
√1 sinh( −br) ifb<0.
−b
√
Notethatforb>0thefunctionw (r)admitsasmoothextensiontor =π/ b.
b
Proposition3.4(See[O’N83,GW79,Gri99a]). LetMmbeaw−modelspacewithwarp-
w
ingfunctionw(r)andcentero . Thedistancesphereofradiusrandcentero inMm is
w w w
thefiberπ−1(r). Thisdistancespherehastheconstantmeancurvatureη (r)= w(cid:48)(r) On
w w(r)
theotherhand,theo -radialsectionalcurvaturesofMmateveryx∈π−1(r)(forr >0)
w w
areallidenticalanddeterminedby
w(cid:48)(cid:48)(r)
K (σ )=− .
ow,Mw x w(r)
10 V.GIMENOANDS.MARKVORSEN
Remarkd. Thew−modelspacesarecompletelydeterminedviawbythemeancurvatures
ofthesphericalfibersSw:
r
η (r)=w(cid:48)(r)/w(r) ,
w
bythevolumeofthefiber
Vol(Sw) =V wm−1(r) ,
r 0
andbythevolumeofthecorrespondingball,forwhichthefiberistheboundary
(cid:90) r
Vol(Bw) = V wm−1(t)dt .
r 0
0
HereV denotesthevolumeoftheunitsphereS0,m−1, (wedenoteingeneralasSb,m−1
0 1 r
thesphereofradiusr intherealspaceformKm(b)). Thelattertwofunctionsdefinethe
isoperimetricquotientfunctionasfollows
q (r) = Vol(Bw)/Vol(Sw) .
w r r
Weobservemoreoverthatthefluxofthegeodesicdistancefunctionr fromthecenterto
o
themodelspaceis
(cid:90)
Jw(R)= |∇r|dσ =Vol(Sw) .
r R
Sw
R
Besidesthealreadydefinedcomparisoncontrollersfortheradialsectionalcurvaturesof
Nn,weshallneedtwofurtherpurelyintrinsicconditionsonthemodelspaces:
Definition3.5. Agivenw−modelspace Mm iscalledbalancedfrombelowandbalanced
w
fromabove,respectively,ifthefollowingweightedisoperimetricconditionsaresatisfied:
Balancefrombelow: q (r)η (r) ≥1/m forall r ≥0 ;
w w
Balancefromabove: q (r)η (r) ≤1/(m−1) forall r ≥0 .
w w
Amodelspaceiscalledtotallybalancedifitisbalancedbothfrombelowandfromabove.
3.2. Laplacian comparison for radial functions. Let us recall the expression of the
Laplacianonmodelspacesforradialfunctions
Proposition3.6(See[O’N83],[GW79]and[Gri99a]). LetMn beamodelspace,denote
w
byr : Mn−{o } → R+ thegeodesicdistancefromthecentero ,letf : R → Rbea
w w w
smoothfunction,then
(3.2) ∆Mwn(f ◦r)=f(cid:48)(cid:48)◦r+(n−1)(f(cid:48)·ηw)◦r .
ApplyingtheHessiancomparisontheoremsgivenin[GW79]wecanobtain(see[MP06]
forinstance)
Proposition 3.7. Let ϕ : Pm → Nn be an immersion into a manifold N with a pole.
Suppose the the radial sectional curvatures K of N are bounded from above by the
N
radialsectionalcurvaturesofamodelspaceMmasfollows:
w
w(cid:48)(cid:48)
(3.3) K ≤− ◦r .
N w
Let f : R → R be a smooth function with f(cid:48) ≥ 0, and dennote by r : P → R+ the
extrinsicdistancefunction. Then
∆P (f ◦r)≥|∇Pr|(f(cid:48)(cid:48)−f(cid:48)·η )◦r
w
(3.4)
+m(f(cid:48)·η )◦r+m(cid:104)∇Nr,H (cid:105)f(cid:48)◦r ,
w P
whereH denotesthemeancurvaturevectorofP inN.
P