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1、ChristopherHopperBenAndrewsTheRicciFlowinRiemannianGeometryAcompleteproofofthedifferentiable1=4-pinchingspheretheorem27July2010SpringerAuthorsChristopherHopperMathematicalInstitute24{29StGiles'OxfordOX13LBEnglandhopper@maths.ox.ac.ukBenAndrewsAustralianNationalUniversity
2、CanberraACT0200Australiaben.andrews@anu.edu.auTobepublishedinSpringer'sLectureNotesinMathematicsbookseries.Theoriginalpublicationisavailableatwww.springerlink.com/content/110312/Forintheverytorrent,tempest,andasImaysay,whirlwindofyourpassion,youmustacquireandbegetatempera
3、ncethatmaygiveitsmoothness.
4、Shakespeare,Hamlet.ivPrefaceThereisafamoustheorembyRauch,KlingenbergandBergerwhichstatesthatacompletesimplyconnectedn-dimensionalRiemannianmanifold,forwhichthesectionalcurvaturesarestrictlybetween1and4,ishomeomorphictoan-sphere.Ithasbeenalongst
5、andingopenconjectureastowhetherornotthe`homeomorphism'conclusioncouldbestrengthenedtoa`di eo-morphism'.SincetheintroductionoftheRicci
owbyHamilton[Ham82b]sometwodecadesago,therehavebeenseveralinroadsintothisproblem
6、particularlyindimensionsthreeandfour
7、whichhavethrownlight
8、uponapossibleproofofthisresult.Onlyrecentlyhasthisconjecture(andaconsiderablystrongergeneralisation)beenprovedbySimonBrendleandRichardSchoen.Theaimofthepresentbookistoprovideauni edexpositoryaccountofthedi eren-tiable1=4-pinchingspheretheoremtogetherwiththenecessarybackgr
9、oundmaterialandrecentconvergencetheoryfortheRicci
owinn-dimensions.Thisaccountshouldbeaccessibletoanyonefamiliarwithenoughdi erentialgeometrytofeelcomfortablewithtensors,covariantderivatives,andnormalcoordinates;andenoughanalysistofollowstandardpdearguments.Theproofwepres
10、entisself-contained(exceptforthequotedCheeger-GromovcompactnesstheoremforRiemannianmetrics),andincorporatesseveralim-provementsonwhatiscurrentlyavailableintheliterature.Broadlyspeaking,thestructureofthisbookfallsintothreemaintop-ics.The rstcentresaroundtheintroductionanda
11、nalysistheRicci
owasageometricheat-typepartialdi erentialequation.ThesecondconcernsPerel'man'smo
