佩雷尔曼的论文

arXiv:math/0211159v1[math.DG]11Nov2002TheentropyformulafortheRicciflowanditsgeometricapplicationsGrishaPerelman∗February1,2008Introduction1.TheRicciflowequation,introducedbyRichardHamilton[H1],istheevolutionequationddtgij(t)=−2Rijforariemannianmetricgij(t).Inhisseminalpaper,Hamiltonprovedthatthisequationhasauniquesolutionforashorttimeforanarbitrary(smooth)metriconaclosedmanifold.TheevolutionequationforthemetrictensorimpliestheevolutionequationforthecurvaturetensoroftheformRmt=△Rm+Q,whereQisacertainquadraticexpressionofthecurvatures.Inparticular,thescalarcurvatureRsatisfiesRt=△R+2|Ric|2,sobythemaximumprincipleitsminimumisnon-decreasingalongtheflow.Bydevelopingamaximumprinciplefortensors,Hamilton[H1,H2]provedthatRicciflowpreservesthepositivityoftheRiccitensorindimensionthreeandofthecurvatureoperatorinalldimensions;moreover,theeigenvaluesoftheRiccitensorindimensionthreeandofthecurvatureoperatorindimensionfouraregettingpinchedpoint-wiselyasthecurvatureisgettinglarge.Thisobservationallowedhimtoprovetheconvergenceresults:theevolvingmetrics(onaclosedmanifold)ofpositiveRiccicurvatureindimensionthree,orpositivecurvatureoperator∗St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191011,Russia.Email:[e-mail]@math.sunysb.edu;IwaspartiallysupportedbypersonalsavingsaccumulatedduringmyvisitstotheCourantInstituteintheFallof1992,totheSUNYatStonyBrookintheSpringof1993,andtotheUCatBerkeleyasaMillerFellowin1993-95.I’dliketothankeveryonewhoworkedtomakethoseopportunitiesavailabletome.1indimensionfourconverge,moduloscaling,tometricsofconstantpositivecurvature.WithoutassumptionsoncurvaturethelongtimebehaviorofthemetricevolvingbyRicciflowmaybemorecomplicated.Inparticular,astap-proachessomefinitetimeT,thecurvaturesmaybecomearbitrarilylargeinsomeregionwhilestayingboundedinitscomplement.Insuchacase,itisusefultolookattheblowupofthesolutionfortclosetoTatapointwherecurvatureislarge(thetimeisscaledwiththesamefactorasthemetricten-sor).Hamilton[H9]provedaconvergencetheorem,whichimpliesthatasubsequenceofsuchscalingssmoothlyconverges(modulodiffeomorphisms)toacompletesolutiontotheRicciflowwheneverthecurvaturesofthescaledmetricsareuniformlybounded(onsometimeinterval),andtheirinjectivityradiiattheoriginareboundedawayfromzero;moreover,ifthesizeofthescaledtimeintervalgoestoinfinity,thenthelimitsolutionisancient,thatisdefinedonatimeintervaloftheform(−∞,T).Ingeneralitmaybehardtoanalyzeanarbitraryancientsolution.However,Ivey[I]andHamilton[H4]provedthatindimensionthree,atthepointswherescalarcurvatureislarge,thenegativepartofthecurvaturetensorissmallcomparedtothescalarcurvature,andthereforetheblow-uplimitshavenecessarilynonneg-ativesectionalcurvature.Ontheotherhand,Hamilton[H3]discoveredaremarkablepropertyofsolutionswithnonnegativecurvatureoperatorinar-bitrarydimension,calledadifferentialHarnackinequality,whichallows,inparticular,tocomparethecurvaturesofthesolutionatdifferentpointsanddifferenttimes.TheseresultsleadHamiltontocertainconjecturesonthestructureoftheblow-uplimitsindimensionthree,see[H4,§26];thepresentworkconfirmsthem.Themostnaturalwayofformingasingularityinfinitetimeisbypinchingan(almost)roundcylindricalneck.Inthiscaseitisnaturaltomakeasurgerybycuttingopentheneckandgluingsmallcapstoeachoftheboundaries,andthentocontinuerunningtheRicciflow.TheexactprocedurewasdescribedbyHamilton[H5]inthecaseoffour-manifolds,satisfyingcertaincurvatureassumptions.Healsoexpressedthehopethatasimilarprocedurewouldworkinthethreedimensionalcase,withoutanyaprioryassumptions,andthatafterfinitenumberofsurgeries,theRicciflowwouldexistforalltimet→∞,andbenonsingular,inthesensethatthenormalizedcurvatures˜Rm(x,t)=tRm(x,t)wouldstaybounded.ThetopologyofsuchnonsingularsolutionswasdescribedbyHamilton[H6]totheextentsufficienttomakesurethatnocounterexampletotheThurstongeometrizationconjecturecan2occuramongthem.Thus,theimplementationofHamiltonprogramwouldimplythegeometrizationconjectureforclosedthree-manifolds.InthispaperwecarryoutsomedetailsofHamiltonprogram.Themoretechnicallycomplicatedarguments,relatedtothesurgery,willbediscussedelsewhere.WehavenotbeenabletoconfirmHamilton’shopethattheso-lutionthatexistsforalltimet→∞necessarilyhasboundednormalizedcurvature;stillweareabletoshowthattheregionwherethisdoesnotholdislocallycollapsedwithcurvatureboundedbelow;byourearlier(partlyunpublished)workthisisenoughfortopologicalconclusions.OurpresentworkhasalsosomeapplicationstotheHamilton-Tiancon-jectureconcerningK¨ahler-RicciflowonK¨ahlermanifoldswithpositivefirstChernclass;thesewillbediscussedinaseparatepaper.2.TheRicciflowhasalsobeendiscussedinquantumfieldtheory,asanap-proximationtotherenormalizationgroup(RG)flowforthetwo-dimensionalnonlinearσ-model,see[Gaw,§3]andreferencestherein.Whilemyback-groundinquantumphysicsisinsufficienttodiscussthisonatechnicallevel,IwouldliketospeculateontheWilsonianpictureoftheRGflow.Inthispicture,tcorrespondstothescaleparameter;thelargerist,thelargeristhedistancescaleandthesmalleristheenergyscale;tocomputesomethingonalowerenergyscaleonehastoaverage