Cao-2012-The Kähler-Ricci flow on Fano manifolds

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1、THEKAHLER-RICCIFLOWONFANOMANIFOLDS¨HUAI-DONGCAOContents1.Preliminaries42.TheK¨ahler-RicciflowandthenormalizedK¨ahler-Ricciflow73.TheLi-Yau-HamiltoninequalitiesforKRF144.Perelman’sµ-entropyandκ-noncollapsingtheorems205.UniformcurvatureanddiameterestimatesforNKRFwithnonnegativebisectionalcurvature266.Pe

2、relman’suniformscalarcurvatureanddiameterestimatesforNKRF287.RemarksontheformationofsingularitiesinKRF36References41IntroductionInthislecturenotes,weaimatgivinganintroductiontotheK¨ahler-Ricciflow(KRF)onFanomanifolds,i.e.,compactK¨ahlermanifoldswithpositivefirstChernclass.Itwillcovermostlysomeofthedev

3、elopmentsoftheKRFinitsfirsttwentyyears(1984-2003),especiallyanessentiallyself-containedexpositionofPerelman’suniformestimatesonthescalarcurvature,thediameter,andtheRiccipotentialfunction(inC1-norm)forthenormalizedK¨ahler-Ricciflow(NKRF),includingthemonotonicityofPerelman’sµ-entropyandκ-noncollapsingth

4、eoremsfortheRicciflowoncompactmanifolds.ExceptinthelastsectionwhereweshallbrieflydiscusstheformationofsingularitiesoftheKRFinFanocase,muchoftherecentprogresssincePerelman’suniformestimatesarenottouchedhere,especiallythosebyPhong-Sturm[59]andPhong-Song-Sturm-Weinkove[60,61,62](seealso[54,arXiv:1212.622

5、7v1[math.DG]26Dec201218,73,79,51,86]etc.)tyingtheconvergenceoftheNKRFtoanotionofGITstabilityforthediffeomorphismgroup,inthespiritoftheconjectureofYau[85](seealso[74,30]).Wehopetodiscussthesedevelopments,aswellasmanyworksrelatedtoK¨ahler-Riccisolitons,onanotheroccasion.Wealsoreferthereaderstotherecent

6、lecturenotesbyJ.SongandB.Weinkove[71]forsomeoftheothersignificantdevelopmentsinKRF.Inspring1982,YauinvitedRichardHamiltontogiveatalkattheInstituteforAdvancedStudy(IAS)onhisnewlycompletedseminalwork“Three-manifoldswithpositiveRiccicurvature”[36].Shortlyafter,Yauaskedme,BenChowandNgaimingMoktopresentHa

7、milton’sworkontheRicciflowindetailsatYau’sIASgeometryseminar.Atthetime,BenChowandIwerefirstyeargraduatestudents,andMokwasaninstructoratPrincetonUniversity.Therewasanotherfellowfirst1PartiallysupportedbyN