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1、FiniteextinctiontimeforthesolutionstotheRicciflowoncertainthree-manifoldsGrishaPerelman∗February1,2008InourpreviouspaperweconstructedcompletesolutionstotheRicciflowwithsurgeryforarbitraryinitialriemannianmetricona(closed,oriented)three-manifold[P,6.1],andusedthebehaviorofsuchsolutionstoclas
2、sifythree-manifoldsintothreetypes[P,8.2].Inparticular,thefirsttypeconsistedofthosemanifolds,whoseprimefactorsarediffeomorphiccopiesofsphericalspaceformsandS2×S1;theywerecharacterizedbythepropertythattheyadmitmetrics,thatgiverisetosolutionstotheRicciflowwithsurgery,whichbecomeextinctinfiniteti
3、me.Whilethisclassificationwassufficienttoanswertopologicalques-tions,ananalyticalquestionofsignificantindependentinterestremainedopen,namely,whetherthesolutionbecomesextinctinfinitetimeforeveryinitialmetriconamanifoldofthistype.Inthisnoteweprovethatthisisindeedthecase.Ourargument(incon-junctio
4、nwith[P,§1-5])alsogivesadirectproofofthesocalled”elliptizationconjecture”.Itturnsoutthatitdoesnotrequireanysubstantiallynewideas:weuseonlyaversionoftheleastareadiskargumentfrom[H,§11]andaregu-larizationofthecurveshorteningflowfrom[A-G].1Finitetimeextinction1.1Theorem.LetMbeaclosedorientedt
5、hree-manifold,whoseprimedecom-positioncontainsnoasphericalfactors.ThenforanyinitialmetriconMthesolutiontotheRicciflowwithsurgerybecomesextinctinfinitetime.arXiv:math/0307245v1[math.DG]17Jul2003ProofforirreducibleM.LetΛMdenotethespaceofallcontractibleloopsinC1(S1→M).Givenariemannianmetricgon
6、Mandc∈ΛM,defineA(c,g)tobetheinfimumoftheareasofalllipschitzmapsfromD2toM,whoserestrictionto∂D2=S1isc.ForafamilyΓ⊂ΛMletA(Γ,g)bethesupremumofA(c,g)overallc∈Γ.Finally,foranontrivialhomotopyclassα∈π∗(ΛM,M)letA(α,g)betheinfimumofA(Γ,g)overallΓ∈α.SinceMisnotaspherical,itfollowsfromaclassical(andel
7、ementary)resultofSerrethatsuchanontrivialhomotopyclassexists.∗St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191023,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu11.2Lemma.(cf.[H,§11])IfgtisasmoothsolutiontotheRicciflow,thenfor
