Lecture notes on the K&amp#168;ahler-Ricci flow1.pdf

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1、LecturenotesontheK¨ahler-Ricciflow1JianSong∗andBenWeinkove†November21,2011ContentsIntroduction.......................................31Preliminaries61.1K¨ahlermanifolds.................................61.2Normalcoordinates................................71.3Curvature....................

2、.................81.4Themaximumprinciple.............................101.5Otheranalyticresultsanddefinitions......................121.6Dolbeaultcohomology,linebundlesanddivisors...............131.7Notionsofpositivityoflinebundles......................152GeneralestimatesfortheK¨ahler-Ric

3、ciflow192.1TheK¨ahler-Ricciflow..............................192.2Evolutionofscalarcurvature..........................192.3Evolutionofthetraceofthemetric.......................212.4TheparabolicSchwarzLemma.........................222.5The3rdorderestimate.............................242.6

4、Curvatureandhigherderivativebounds....................273MaximalexistencetimefortheK¨ahler-Ricciflow323.1TheparabolicMonge-Amp`ereequation.....................323.2Estimatesforthepotentialandthevolumeform...............343.3Auniformboundfortheevolvingmetric....................354Converg

5、enceoftheflowinthecasesc1(M)<0andc1(M)=0384.1ThenormalizedK¨ahler-Ricciflowwhenc1(M)<0..............384.2Thecaseofc1(M)=0:Yau’szerothorderestimate.............414.3Thecaseofc1(M)=0:higherorderestimatesandconvergence.......441Thefirst-namedauthorwassupportedinpartbyanNSFCAREERgrantDMS-08-4

6、7524andthesecond-namedauthorbytheNSFgrantsDMS-08-48193andDMS-11-05373.Bothauthorswerealsosup-portedinpartbySloanResearchFellowships.15ThecasewhenKMisbigandnef485.1Smoothminimalmodelsofgeneraltype....................485.2Estimates.....................................495.3Convergenceofth

7、eflowanduniquenessofthelimit.............555.4Furtherestimatesusingpluripotentialtheory.................576TheK¨ahler-Ricciflowonaproductellipticsurface606.1EllipticsurfacesandtheK¨ahler-Ricciflow...................606.2Estimates.....................................626.3Fibercollapsinga