supremum of Peleman's entropy and Kahler-Ricci flow on a Fano maniflod英文学习材料

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1、SUPREMUMOFPERELMAN’SENTROPYANDKAHLER-RICCIFLOWONAFANOMANIFOLD¨∗∗∗GANGTIAN,SHIJINZHANG,ZHENLEIZHANG,ANDXIAOHUAZHUAbstract.Inthispaper,weextendthemethodin[TZhu5]tostudytheenergylevelL(·)ofPerelman’sentropyλ(·)forK¨ahler-RicciflowonaFanomanifold.Consequently,wefirstcomputethesupremumofλ(·)inK¨ahlerclass

2、2πc1(M)underanassumptionthatthemodifiedMabuchi’sK-energyµ(·)definedin[TZhu2]isboundedfrombelow.Secondly,wegiveanalternativeprooftothemaintheoremabouttheconvergenceofK¨ahler-Ricciflowin[TZhu3].IntroductionInthispaper,weextendthemethodin[TZhu5]tostudytheenergylevelL(·)ofPerelman’sentropyλ(·)forK¨ahler-R

3、icciflowonann-dimensionalcompactK¨ahlermanifold(M,J)withpositivefirstChernclassc1(M)>0(namelycalledaFanomanifold).WewillshowthatL(·)isindependentofchoiceofinitialK¨ahlermetricsin2πc1(M)underanassumptionthatthemodifiedMabuchi’sK-energyµ(·)isboundedfrombelow(cf.Proposition3.1inSection3).ThemodifiedMabuch

4、i’sK-energyµ(·)isageneralizationofMabuchi’sK-energy.Itwasshowedin[TZhu2]thatµ(·)isboundedfrombelowifMadmitsaK¨ahler-Riccisoliton.AsanapplicationofProposition3.1,wefirstcomputethesupremumofPerelman’sentropyλ(·)inK¨ahlerclass2πc1(M)[Pe].Moreprecisely,weprovethatarXiv:1107.4018v1[math.DG]20Jul2011Theor

5、em0.1.SupposethatthemodifiedMabuchi’sK-energyisboundedfrombelow.Then′′−n(0.1)sup{λ(g)

6、g∈KX}=(2π)[nV−NX(c1(M))].HerethequantityNX(c1(M))isanonnegativeinvarianceinKXanditiszeroifftheFutaki-invariantvanishes[Fu].WedenoteKXtobeaclass1991MathematicsSubjectClassification.Primary:53C25;Secondary:53C55,58E11.

7、Keywordsandphrases.K¨ahler-Ricciflow,K¨ahler-Riccisolitons,Perelman’sentropy.*PartiallysupportedbyagrantofBMCE11224010007inChina.**PartiallysupportedbytheNSFCGrant10990013.1∗∗∗2GANGTIAN,SHIJINZHANG,ZHENLEIZHANG,ANDXIAOHUAZHUofKX-invariantK¨ahlermetricsin2πc1(M),whereKXisanone-parametercompactsubgrou

8、pofholomorphismstransformationgroupgeneratedbyanextremalholomorphicvectorfieldXforK¨ahler-RiccisolitonsonM[TZhu2].WenotethatwedonotneedtoassumeanexistenceofK¨ahler-RiccisolitonsinTheorem0.1.Infact,ifweassume