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1、3ClayMathematicsMonographsForover100yearsthePoincaréConjecture,whichproposesatopo-Volume3logicalcharacterizationofthe3-sphere,hasbeenthecentralquestionintopology.Sinceitsformulation,ithasbeenrepeatedlyattacked,withoutRicciFlowandthePoincaréConjecturesuccess,usingvarioustopo
2、logicalmethods.ItsimportanceanddifficultywerehighlightedwhenitwaschosenasoneoftheClayMathematicsInstitute'ssevenMillenniumPrizeProblems.In2002and2003GrigoryPerelmanpostedthreepreprintsshowinghowtousegeometricargu-ments,inparticulartheRicciflowasintroducedandstudiedbyHamilto
3、n,RicciFlowandthetoestablishthePoincaréConjectureintheaffirmative.ThisbookprovidesfulldetailsofacompleteproofofthePoincaréConjecturefollowingPerelman'sthreepreprints.Afteralengthyintro-ductionthatoutlinestheentireargument,thebookisdividedintofourPoincaréConjectureparts.Thef
4、irstpartreviewsnecessaryresultsfromRiemanniangeometryandRicciflow,includingmuchofHamilton'swork.ThesecondpartstartswithPerelman'slengthfunction,whichisusedtoestablishcrucialnon-collapsingtheorems.Thenitdiscussestheclassificationofnon-collapsed,ancientsolutionstotheRicciflow
5、equation.ThethirdpartconcernstheexistenceofRicciflowwithsurgeryforallpositivetimeandananalysisofthetopologicalandgeometricchangesintroducedbysurgery.ThelastpartfollowsPerelman'sthirdpreprinttoprovethatwhentheinitialRiemannian3-manifoldhasfinitefundamentalgroup,Ricciflowwith
6、surgerybecomesextinctafterfinitetime.TheproofsofthePoincaréConjectureandthecloselyrelated3-dimensionalsphericalspace-formconjecturearethenimmediate.TheexistenceofRicciflowwithsurgeryhasapplicationto3-manifoldsfarbeyondthePoincaréConjecture.ItformstheheartoftheproofviaRiccif
7、lowofThurston'sGeometrizationConjecture.Thurston'sGeometrizationConjecture,whichclassifiesallcompact3-manifolds,willbethesubjectofafollow-uparticle.TheorganizationofthematerialinthisbookdiffersfromthatgivenbyPerelman.Fromthebeginningtheauthorspresentallanalyticandgeometrica
8、rgumentsinthecontextofRicciflowwithsurgery.Inaddition,thefourthpartisamuch-expande
