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1、TheentropyformulafortheRicciflowanditsgeometricapplicationsGrishaPerelman∗February1,2008Introduction1.TheRicciflowequation,introducedbyRichardHamilton[H1],istheevolutionequationdg(t)=−2Rforariemannianmetricg(t).Inhisdtijijijseminalpaper,Hamiltonprovedthatthisequationhasauniquesolutionforasho
2、rttimeforanarbitrary(smooth)metriconaclosedmanifold.TheevolutionequationforthemetrictensorimpliestheevolutionequationforthecurvaturetensoroftheformRmt=△Rm+Q,whereQisacertainquadraticexpressionofthecurvatures.Inparticular,thescalarcurvatureRsatisfiesR=△R+2
3、Ric
4、2,sobythemaximumprincipleitsmin
5、imumtisnon-decreasingalongtheflow.Bydevelopingamaximumprinciplefortensors,Hamilton[H1,H2]provedthatRicciflowpreservesthepositivityoftheRiccitensorindimensionthreeandofthecurvatureoperatorinalldimensions;moreover,theeigenvaluesoftheRiccitensorindimensionthreeandofthecurvatureoperatorindimensi
6、onfouraregettingpinchedpoint-arXiv:math/0211159v1[math.DG]11Nov2002wiselyasthecurvatureisgettinglarge.Thisobservationallowedhimtoprovetheconvergenceresults:theevolvingmetrics(onaclosedmanifold)ofpositiveRiccicurvatureindimensionthree,orpositivecurvatureoperator∗St.PetersburgbranchofSteklov
7、MathematicalInstitute,Fontanka27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu;IwaspartiallysupportedbypersonalsavingsaccumulatedduringmyvisitstotheCourantInstituteintheFallof1992,totheSUNYatStonyBrookintheSpringof1993,andtotheUCatBerkeleyasaMillerFellowin
8、1993-95.I’dliketothankeveryonewhoworkedtomakethoseopportunitiesavailabletome.1indimensionfourconverge,moduloscaling,tometricsofconstantpositivecurvature.WithoutassumptionsoncurvaturethelongtimebehaviorofthemetricevolvingbyRicciflowmaybemorecomplicated.Inparticular,astap-proachessomefinitetim
9、eT,thecurvaturesmaybecomearbitrarilylargeinsomeregionwhilestayingboundedinitscomplement.Insuchacase,itisusefultolookattheblowupofthesolutionfortclosetoTatapointwherecurvatureislarge(thetimeisscaledwiththesamefactorasthemetricten-sor).Hamilton[H9]provedaconverg
