Non-trivial bounded harmonic functions on Cartan-Hadamard manifolds of unbounded curvature [Elektronische Ressource] / vorgelegt von Stefanie Ulsamer

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Non-trivial Bounded HarmonicFunctions on Cartan-HadamardManifolds of UnboundedCurvatureDissertation zur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.) derNaturwissenschaftlichen Fakult at I { Mathematikder Universit at Regensburgvorgelegt vonStefanie Ulsameraus RegensburgRegensburg, im Oktober 2003Promotionsgesuch eingereicht am: 15. Oktober 2003Diese Arbeit wurde angeleitet von: Prof. Dr. Anton ThalmaierPrufungsaussc huss: Vorsitzender: Prof. Dr. Gun ter Tamme1. Gutachter: Prof. Dr. Anton Thalmaier2.hter: Prof. Dr. Marc ArnaudonErsatzprufer: Prof. Dr. Wolfgang HackenbrochWeiterer Prufer: Prof. Dr. Knut KnorrTermin der mundlic hen Prufung: 19. Dezember 2003ContentsIntroduction 51 Some Background on Di eren tial Geometry 151.1 Fundamentals and De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Cartan-Hadamard Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3 The Sphere at In nit y and the Dirichlet Problem at In nit y . . . . . . . . . 182 Brownian Motion on Riemannian Manifolds 212.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Important Properties of Brownian Motion . . . . . . . . . . . . . . . . . . . 222.3 Brownian Motion and Harmonic Functions . . . . . . . . . . . . . . . . . . 232.4 The Martin Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sujets

• Mathematics

Informations

• Publié par: universitat_regensburg
• Publié le: 01 janvier 2004
• Nombre de lectures: 22

Extrait

Non-trivial

Bounded

Harmonic

Cartan-HadamardonunctionsF

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DoktorgradesdesErlangungzurDissertation

derNaturwissenschaften(Dr.rer.nat.)der

NaturwissenschaftlichenFakult¨atI–Mathematik

der

Regensburgatersit¨Univ

vorgelegtvon

UlsamerStefanie

Regensburgaus

erOktobimRegensburg,

2003

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Pr¨ufungsausschuss:Vorsitzender:
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Terminderm¨undlichenPr¨ufung:

15.Prof.OktobDr.erAnton2003Thalmaier

Prof.Dr.G¨unterTamme
ThalmaiertonAnDr.Prof.ArnaudonMarcDr.Prof.Prof.Dr.WolfgangHackenbroch
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19.2003erbDezem

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1SomeBackgroundonDifferentialGeometry15
1.1FundamentalsandDefinitions..........................15
1.2Cartan-HadamardManifolds..........................17
1.3TheSphereatInfinityandtheDirichletProblematInfinity.........18

2BrownianMotiononRiemannianManifolds21
2.1Definitions.....................................21
2.2ImportantPropertiesofBrownianMotion...................22
2.3BrownianMotionandHarmonicFunctions..................23
2.4TheMartinBoundary..............................30

3ANon-LiouvilleManifoldwithDegenerateAngularBehaviourofBM36
3.1ComputationoftheSectionalCurvature....................37
3.2TheSphereatInfinityS∞(M).........................39
3.3PropertiesoftheFunctiong...........................39
3.4ConstructionoftheFunctiong.........................41
3.5BrownianMotiononM.............................43
3.6Non-TrivialShift-InvariantEventsforB....................58
3.7ConstructionoftheFunctionq.........................70
3.8GeometricInterpretationoftheAsymptoticBehaviourofBrownianMotion71

4FurtherConstructionsofNon-LiouvilleManifoldsofUnboundedCur-
77aturev4.1TheManifoldofAncona.............................78
4.2PropertiesandConstructionoftheFunctionh................79
4.3TheoremAofAncona..............................82
4.4TheoremAExtendedtoHigherDimensions..................84
4.5FurtherConstructionsandConsiderations...................87
4.6SomeConcludingRemarks...........................93

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Thequestionunderwhichconditionsthereexistnon-trivialboundedharmonicfunctions
onRiemannianmanifolds(M,g)hasbeenofgreatinteresttomanymathematiciansand
is.stillAfunctionh:M→RiscalledharmonicifitisasmoothsolutionoftheLaplaceequation
ΔMu=0,(1)
whereΔMistheLaplace-BeltramioperatorontheRiemannianmanifoldM.
Ithasbeenknownsince1957,see[Hu],thattherearenonon-constantboundedharmonic
functionsonacompletesurface,i.e.completeRiemannianmanifoldofdimensiontwo,
withpositivecurvature.Ontheotherhand,itfollowsfromtheAhlfors-SchwarzLemma,
[Ah],thatasimplyconnectedsurfacewithcurvatureboundedfromabovebyanegative
constantisconformallyequivalenttotheunitdiscandconsequentlypossessesnon-trivial
functions.harmonicoundedbHenceitisanaturalquestiontoask,whethercurvature–sectionalcurvature,tobe
moreprecise–isagoodcriterioninalldimensionstodistinguishbetweenRiemannian
manifoldswhichadmitnon-trivialboundedharmonicfunctionsandso-calledLiouville
manifolds,i.e.Riemannianmanifoldswhereconstantfunctionsaretheonlysolutionsto
(1).equationAsanimmediateconsequenceoftheinfinitesimalversionoftheHarnackinequalityproven
byYauin[Y],Theorem3”,everypositiveandthereforeeveryboundedharmonicfunction
onacompleteRiemannianmanifold(ofarbitrarydimension)withnon-negative(i.e.≥0)
t.constanisaturecurvIncaseofaCartan-Hadamardmanifold,i.e.acompletesimplyconnectedRiemannian
manifoldwithnon-positive(i.e.≤0)sectionalcurvature,thereisthefollowingconjecture
ofGreeneandWu,thatwas(inaslightlyrelaxedversion)alsoaconsiderationofDynkin
in[D1].Inthefollowing,r(x)denotestheradialpartofx∈M:
Conjecture0.1(cf.[G-W]and[H-M],p.767).
Let(M,g)beaCartan-Hadamardmanifoldwithsectionalcurvatures
SectxM≤−cr(x)−2
forsomeconstantcandallx∈Minthecomplementofacompactset.Thenthereare
non-constantboundedharmonicfunctionsonM.

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Uptonowthereisnoproofknown,butthereareseveralaffirmativeresultsinthisdirection.
Wearegoingtogiveashorthistoricaloverviewoftheseresultsandthemethodsusedfor
ofs:protheirForaCartan-HadamardmanifoldMofdimensiondthereisanaturalgeometricboundary,
thesphereatinfinityS∞(M),suchthatM∪S∞(M)equippedwiththeconetopologyis
homeomorphictotheunitballB⊂Rdwithboundary∂B=Sd−1,cf.[E-O´N],[B-O´N]
and[Kl].Usingpolarcoordinates(r,ϑ)forM,asequence(rn,ϑn)n∈NofpointsinM
convergestoapointofS∞(M)ifandonlyifrn→∞andϑn→:ϑ.
Givenacontinuousfunctionf:S∞(M)→RtheDirichletproblematinfinityistofinda
harmonicfunctionh:M→RwhichextendscontinuouslytoS∞(M)andtherecoincides
withthegivenfunctionf.TheDirichletproblematinfinityiscalledsolvableifthisis
possibleforeverysuchfunctionf.Hencethequestionwhetherthereexistnon-trivial
boundedharmonicfunctionsonMisnaturallyrelatedtothequestioniftheDirichlet
problematinfinityforMissolvable.
In1983,AndersonprovedthattheDirichletproblematinfinityisuniquelysolvablefor
Cartan-Hadamardmanifoldswithpinchednegativecurvature,i.e.forcompletesimply
connectedRiemannianmanifoldsMwhosesectionalcurvaturessatisfy
−a2≤SectxM≤−b2forallx∈M,
wherea2>b2>0arearbitraryconstants.See[An],Theorem3.2.Themainideaof
theproofwastousebarrierfunctionsandPerron’smethodtoobtainthedesiredresults.
EssentiallythesameideasareusedbyChoiin1984toshowthatincaseofamodel
manifold(M,g)theDirichletproblematinfinityissolvableiftheradialcurvatureis
boundedfromaboveby−A/(r2log(r)).HerebyaRiemannianmanifold(M,g)iscalled
modelifitpossessesapolep∈Mandeverylinearisometryϕ:TpM→TpMcanbe
realizedasthedifferentialofanisometryΦ:M→MwithΦ(p)=p,see[C],Theorem3.6.
Choifurthermoreprovidesacriterion,theconvexconicneighbourhoodcondition,which
yieldssolvabilityoftheDirichletproblematinfinity.
Definition0.2(cf.[C],Definition4.6).
LetMbeaCartan-Hadamardmanifold.Msatisfiestheconvexconicneighbourhood
conditionatx∈S∞(M)ifforanyy∈S∞(M),y=x,thereexistVxandVy⊂M∪S∞(M)
suchthatVxandVyaredisjoin2topensetsofM∪S∞(M)intermsoftheconetopology
andVx∩MisconvexwithC-boundary.Ifthisconditionissatisfiedforallx∈S∞(M),
wesaythatMsatisfiestheconvexconicneighbourhoodcondition.
Dueto[C],Theorem4.7,theDirichletproblematinfinityissolvableforaCartan-
HadamardmanifoldMwithsectionalcurvatureboundedfromaboveby−c2,forc>0,
thatsatisfiestheconvexconicneighbourhoodcondition.
AnotherapproachtotheDirichletproblematinfinityisgivenfromprobabilisticmethods
asitiswellknownthatharmonicfunctionsonaRiemannianmanifoldarecharacterized
bythemeanvaluepropertyforgeodesicballs,seeTheorem2.9.Thispropertyextends
undercertainconditionstothesphereatinfinity,i.e.iftheDirichletproblematinfinity

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forMissolvableandalmostsurelyBζ:=limt→ζBtexistsinS∞(M),where(Bt)t<ζisa
BrownianmotiononMwithlifetimeζ,theuniquesolutionh:M→RtotheDirichlet
problematinfinitywithboundaryfunctionfisgivenas
h(x)=Ef◦Bζxx.(2)
HereBxisaBrownianmotionstartinginx∈M.
Onthecontrary,consideringaBrownianmotiononMsuchthatalmostsurelylimt→ζxBtx
existsinS∞(M)forallx∈M,onecandefinetheharmonicmeasureµxonS∞(M),where
foraBorelsetU⊂S∞(M)
µx(U):=PBζxx∈U.(3)
ForeveryBorelsetU⊂S∞(M)theassignment
x→µx(U)
definesaboundedharmonicfunctionhUonM.Usingthemaximumprincipleforharmonic
functionsitfollowsthathUiseitheridenticallyequalto0or1ortakesvaluesin(0,1).
Furthermore,alltheharmonicmeasuresµxonS∞(M)areequivalent.Showingthatthe
harmonicmeasureclassonS∞(M)isnon-trivialsolvestheDirichletproblematinfinity
forMastheuniquesolutionforagivencontinuousboundaryfunctionf:S∞(M)→Ris
formtheinengivh(x)=S∞(M)f(y)µx(dy).(4)
ThisexplainswhystudyingtheasymptoticbehaviourofBrownianmotiononMisa
convenientmethodtodecidewhethertheDirichletproblemforMissolvableornot.
ThefirstresultsinthisdirectionhavebeenobtainedbyPratbetween1971and1975
(see[P1]and[P2]).HeprovedthatonaCartan-Hadamardmanifoldwherethesectional
curvatureisboundedfromabovebyanegativeconstant−k2,k>0,Brownianmotionis
transient,i.e.almostsurelyallpathsoftheBrownianmotionexitfromMatthesphere
atinfinity([P2],Th´eor`eme1).Ifinadditionthesectionalcurvaturesareboundedfrom
belowbyaconstant−K2,K>k,heshowsthattheangularpartϑ(Bt)oftheBrownian
motionalmostsurelyconvergeswhent→ζ([P2],Th´eor`eme2).Thisisthereasonwhyit
makessensetoconsiderharmonicmeasuresonS∞(M)inthissituation.
In1976,Kiferpresentedastochasticproof,see[K1],Theorem2,thatonCartan-Hadamard
manifoldswithsectionalcurvatureboundedbetweentwonegativeconstantsandsatisfying
acertainadditionalcondition(Condition1in[K1])theDirichletproblematinfinitycan
beuniquelysolved.However,theprooftherewasmerelygiveninexplicittermsforthe
twodimensionalcase.ThecaseofaCartan-Hadamardmanifold(M,g)withpinched
curvaturewithoutadditionalconditionsandarbitrarydimensionwasfinallytreatedby
Kiferin1984inamoreaccurateversionin[K2],Section3.
IndependentlyofAnderson,in1983,Sullivanpresentedastochasticproofofthefactthat
onaCartan-HadamardmanifoldwithpinchedcurvaturetheDirichletproblematinfinity

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isuniquelysolvable(see[S],Theorem1).Thecrucialpointhasbeentoprovethatthe
harmonicmeasureclassisnon-trivialinthiscase.Heobtainshisresultasacorollaryof
theorem:wingfollothe2).Theorem([S],0.3TheoremTheharmonicmeasureclassonS∞(M)=∂(M∪S∞(M))ispositiveoneachnonvoidopen