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1、TheentropyformulafortheRicciflowanditsgeometricapplicationsGrishaPerelman∗February1,2008Introduction1.TheRicciflowequation,introducedbyRichardHamilton[H1],istheevolutionequationdg(t)=−2Rforariemannianmetricg(t).Inhisdtijijijseminalpaper,Hamiltonprovedthatthisequationhasauniquesoluti
2、onforashorttimeforanarbitrary(smooth)metriconaclosedmanifold.TheevolutionequationforthemetrictensorimpliestheevolutionequationforthecurvaturetensoroftheformRmt=△Rm+Q,whereQisacertainquadraticexpressionofthecurvatures.Inparticular,thescalarcurvatureRsatisfiesR=△R+2
3、Ric
4、2,sobythemaxi
5、mumprincipleitsminimumtisnon-decreasingalongtheflow.Bydevelopingamaximumprinciplefortensors,Hamilton[H1,H2]provedthatRicciflowpreservesthepositivityoftheRiccitensorindimensionthreeandofthecurvatureoperatorinalldimensions;moreover,theeigenvaluesoftheRiccitensorindimensionthreeandofth
6、ecurvatureoperatorindimensionfouraregettingpinchedpoint-arXiv:math/0211159v1[math.DG]11Nov2002wiselyasthecurvatureisgettinglarge.Thisobservationallowedhimtoprovetheconvergenceresults:theevolvingmetrics(onaclosedmanifold)ofpositiveRiccicurvatureindimensionthree,orpositivecurvatureo
7、perator∗St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu;IwaspartiallysupportedbypersonalsavingsaccumulatedduringmyvisitstotheCourantInstituteintheFallof1992,totheSUNYatStonyBrookintheSprin
8、gof1993,andtotheUCatBerkeleyasaMillerFellowin1993-95.I’dliketothankeveryonewhoworkedtomakethoseopportunitiesavailabletome.1indimensionfourconverge,moduloscaling,tometricsofconstantpositivecurvature.WithoutassumptionsoncurvaturethelongtimebehaviorofthemetricevolvingbyRicciflowmaybem
9、orecomplicated.Inparticular,astap-proachessomefinitetimeT,thecurvaturesmaybecomearbitrarilylargeinsomeregionwhilestayingboundedinitscomplement.Insuchacase,itisusefultolookattheblowupofthesolutionfortclosetoTatapointwherecurvatureislarge(thetimeisscaledwiththesamefactorasthemetricte
10、n-sor).Hamilton[H9]provedaconverg
