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1、FiniteextinctiontimeforthesolutionstotheRicciflowoncertainthree-manifoldsGrishaPerelman∗February1,2008InourpreviouspaperweconstructedcompletesolutionstotheRicciflowwithsurgeryforarbitraryinitialriemannianmetricona(closed,oriented)three-manifold[P,6.1],andusedth
2、ebehaviorofsuchsolutionstoclassifythree-manifoldsintothreetypes[P,8.2].Inparticular,thefirsttypeconsistedofthosemanifolds,whoseprimefactorsarediffeomorphiccopiesofsphericalspaceformsandS2×S1;theywerecharacterizedbythepropertythattheyadmitmetrics,thatgiverisetos
3、olutionstotheRicciflowwithsurgery,whichbecomeextinctinfinitetime.Whilethisclassificationwassufficienttoanswertopologicalques-tions,ananalyticalquestionofsignificantindependentinterestremainedopen,namely,whetherthesolutionbecomesextinctinfinitetimeforeveryinitialmetr
4、iconamanifoldofthistype.Inthisnoteweprovethatthisisindeedthecase.Ourargument(incon-junctionwith[P,§1-5])alsogivesadirectproofofthesocalled”elliptizationconjecture”.Itturnsoutthatitdoesnotrequireanysubstantiallynewideas:weuseonlyaversionoftheleastareadiskargum
5、entfrom[H,§11]andaregu-larizationofthecurveshorteningflowfrom[A-G].1Finitetimeextinction1.1Theorem.LetMbeaclosedorientedthree-manifold,whoseprimedecom-positioncontainsnoasphericalfactors.ThenforanyinitialmetriconMthesolutiontotheRicciflowwithsurgerybecomesextin
6、ctinfinitetime.arXiv:math/0307245v1[math.DG]17Jul2003ProofforirreducibleM.LetΛMdenotethespaceofallcontractibleloopsinC1(S1→M).GivenariemannianmetricgonMandc∈ΛM,defineA(c,g)tobetheinfimumoftheareasofalllipschitzmapsfromD2toM,whoserestrictionto∂D2=S1isc.Forafamily
7、Γ⊂ΛMletA(Γ,g)bethesupremumofA(c,g)overallc∈Γ.Finally,foranontrivialhomotopyclassα∈π∗(ΛM,M)letA(α,g)betheinfimumofA(Γ,g)overallΓ∈α.SinceMisnotaspherical,itfollowsfromaclassical(andelementary)resultofSerrethatsuchanontrivialhomotopyclassexists.∗St.Petersburgbran
8、chofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191023,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu11.2Lemma.(cf.[H,§11])IfgtisasmoothsolutiontotheRicciflow,thenfor
