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1、RicciflowonK¨ahlermanifoldsX.X.ChenandG.TianJuly,26,20001IntroductionandmaintheoremsInthelasttwodecades,theRicciflow,introducedbyR.Hamiltonin[7],hasbeenasubjectofintensestudy.TheRicciflowprovidesanindispensabletoolofdeformingRiemannianmetricstowardscanonicalme
2、trics,suchasEinsteinones.Itishopedthatbydeformingametrictoacanonicalmetric,onecanfurtherunderstandgeometricandtopologicalstructuresofunderlyingmanifolds.Forinstance,itwasproved[7]thatanyclosed3-manifoldofpositiveRiccicurvatureisdiffeomorphictoasphericalspace
3、form.Wereferthereadersto[10]formoreinformation.IftheunderlyingmanifoldisaK¨ahlermanifold,thenormalizedRicciflowinacanonicalK¨ahlerclass1preservestheK¨ahlerclass.ItfollowsthattheRicciflowcanbereducedtoafullynonlinearparabolicequationonalmostpluri-subharmonicfu
4、nctions: !n∂ϕω+∂∂ϕ¯=logdet+ϕ−hω,∂tωnwhereϕistheevolvedK¨ahlerpotential;andωisthefixedK¨ahlermetricinthecanonicalK¨ahlerclass,whileRic(ω)isthecorrespondingRicciformandZRic(ω)−ω=∂∂h¯,and(ehω−1)ωn=0.ωMarXiv:math/0010007v1[math.DG]2Oct2000Usually,thisreducedflow
5、iscalledtheK¨ahlerRicciflow.H.D.Cao[2]provedthattheK¨ahlerRicciflowalwayshasaglobalsolution.HealsoprovedthatthesolutionconvergestoaK¨ahler-EinsteinmetricifthefirstChernclassoftheunderlyingK¨ahlermanifoldiszeroornegative.Consequently,hereprovedthefamousCalabi-Y
6、autheorem[17].Ontheotherhand,ifthefirstChernclassoftheunderlyingK¨ahlermanifoldispositive,thesolutionofaK¨ahlerRicciflowmaynotconvergetoanyK¨ahler-Einsteinmetric.ThisisbecausetherearecompactK¨ahlermanifoldswithpositivefirstChernclasswhichdonotadmitanyK¨ahler-E
7、insteinmetrics(cf.[6],[15]).AnaturalandchallengingproblemiswhetherornottheK¨ahlerRicciflowonacompactK¨ahler-Einstein1AK¨ahlerclassiscanonicalifthefirstChernclassisproportionaltothisK¨ahlerclass.1manifoldconvergestoaK¨ahler-Einsteinmetric.ItwasprovedbyS.Bando[
8、1]for3-dimensionalK¨ahlermanifoldsandbyN.Mok[12]forhigherdimensionalK¨ahlermanifoldsthatthepositivityofbisectionalcurvatureispreservedundertheK¨ahlerRicciflow.AlongstandingprobleminthestudyoftheRicciflow
