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1、FiniteextinctiontimeforthesolutionstotheRicci?owoncertainthree-manifoldsGrishaPerelman?February1,2008InourpreviouspaperweconstructedcompletesolutionstotheRicci?owwithsurgeryforarbitraryinitialriemannianmetricona(closed,oriented)three-manifold[P,6.1],andusedthebehavior
2、ofsuchsolutionstoclassifythree-manifoldsintothreetypes[P,8.2].Inparticular,the?rsttypeconsistedofthosemanifolds,whoseprimefactorsaredi?eomorphiccopiesofsphericalspaceformsandS2×S1;theywerecharacterizedbythepropertythattheyadmitmetrics,thatgiverisetosolutionstotheRicci
3、?owwithsurgery,whichbecomeextinctin?nitetime.Whilethisclassi?cationwassu?cienttoanswertopologicalques-tions,ananalyticalquestionofsigni?cantindependentinterestremainedopen,namely,whetherthesolutionbecomesextinctin?nitetimeforeveryinitialmetriconamanifoldofthistype.Int
4、hisnoteweprovethatthisisindeedthecase.Ourargument(incon-junctionwith[P,§1-5])alsogivesadirectproofofthesocalled”elliptizationconjecture”.Itturnsoutthatitdoesnotrequireanysubstantiallynewideas:weuseonlyaversionoftheleastareadiskargumentfrom[H,§11]andaregu-larizationoft
5、hecurveshortening?owfrom[A-G].1Finitetimeextinction1.1Theorem.LetMbeaclosedorientedthree-manifold,whoseprimedecom-positioncontainsnoasphericalfactors.ThenforanyinitialmetriconMthesolutiontotheRicci?owwithsurgerybecomesextinctin?nitetime.arXiv:math/0307245v1[math.DG]17
6、Jul2003ProofforirreducibleM.LetΛMdenotethespaceofallcontractibleloopsinC1(S1→M).GivenariemannianmetricgonMandc∈ΛM,de?neA(c,g)tobethein?mumoftheareasofalllipschitzmapsfromD2toM,whoserestrictionto?D2=S1isc.ForafamilyΓ?ΛMletA(Γ,g)bethesupremumofA(c,g)overallc∈Γ.Finally,f
7、oranontrivialhomotopyclassα∈π?(ΛM,M)letA(α,g)bethein?mumofA(Γ,g)overallΓ∈α.SinceMisnotaspherical,itfollowsfromaclassical(andelementary)resultofSerrethatsuchanontrivialhomotopyclassexists.?St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191023
8、,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu11.2Lemma.(cf.[H,§11])IfgtisasmoothsolutiontotheRicci?ow,thenfor
