Differential.Harnack.Inequality&Ricci.Flow

《Differential.Harnack.Inequality&Ricci.Flow》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ThisisapreprintofthebookpublishedinEMSSeriesLect.Math.(2006)DOI:10.4171/030DierentialHarnackInequalitiesandtheRicciFlowRetoM•ullerNote:Thisisapreliminaryversionofmybook.Inparticular,inthispreprintthePreface,TableofContents,ListofSymbols,IndexandallPictures/Figuresare

2、missing.ThebookcanbeboughtatEMSorAMS.IntroductionInthisbookwecomparethelinearheatequationonastaticmanifoldwiththeRicci ow-whichisanonlinearheatequationforaRiemannianmetricg(t)onM-andwiththeheatequationonamanifoldmovingbyRicci ow.Theaimofthebookistoexplainsomeofthetheo

3、ryofPerelman'srstRicci owpaper[28]aswellasitsthematiccontext.Concretely,wewilldiscussdierentialHarnackinequalities,entropyformulasandPerelman'sL-functional.Inparticular,wewanttoemphasizehowthesethreetopicsarerelatedwitheachotherandwithspecialsolutionsoftheRicci ow,s

4、o-calledRiccisolitons.WewillseethatLi-YautypeHarnackinequalitiesarerelatedwithPerelman'sentropyformulasandtheL-functionalinaquitenaturalway.Tomotivatetheresultsandtostartwithsimplercomputations,wewillrstconsidertheanalogsofthesetopicsfortheheatequationonastaticmanifo

5、ld.WewillseethattheresultsinthestaticandintheRicci owcaseareoftenverysimilarandsometimescoincideonspecialsolutions.Thisintroductioncontainsabriefexpositionofthehistoricalbackgroundandsummarizesthemaincontentofthebook.Hamilton'sRicci owanditsearlysuccessNonlinearheat o

6、wsrstappearedinRiemanniangeometryin1964,whenJamesEellsandJ.H.Sampson[10]introducedtheharmonicmapheat owasatooltodeformgivenmapsu:M!Nbetweentwomanifoldsintoextremalmaps(i.e.criticalpointsinthesenseofcalculusofvariations)fortheenergyfunctionalZ2E(u)=jrujdV:(0.1)MInpart

7、icular,theseso-calledharmonicmapsincludegeodesics,harmonicfunctionsandmin-imalsurfaces.EellsandSampsonshowedthatthegradient owoftheenergyfunctional(0.1)above-theharmonicmapheat ow-willconvergetoaharmonicmapifthetargetmanifoldNhasnegativesectionalcurvature.Sincethen,ge

8、ometricheat owshavebecomeanintensivelystudiedtopicingeometricanalysis.Afundamentalproblemindierentialgeometryistondcanonic