The Canonical Expanding Soliton and Harnack inequalities for Ricci flow

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1、thecanonicalexpandingsolitonandharnackinequalitiesforricciflowEstherCabezas-RivasandPeterM.ToppingJune1,2010AbstractWeintroducethenotionofCanonicalExpandingRicciSoliton,anduseittoderivenewHarnackinequalitiesforRicci ow.ThisviewpointalsogivesgeometricinsightintotheexistingHarnac

2、kinequalitiesofHamiltonandBrendle.1IntroductionRecently,in[4],weintroducedthenotionofCanonicalSoliton.Roughlyspeaking,givenanyRicci owonamanifoldMoveratimeintervalI(1;0),weimaginedthetimedirectionasanadditionalspacedirectionandconstructedashrinkingRiccisolitononMIwithrespect

3、toacompletelynewtimedirection.ConsideringthesesolitonsinthecontextofknownnotionsandtheoremsinRiemanniangeometrytheninducedinterestingconceptsandresultsconcerningtheoriginalRicci ow,manyofwhichwererstdiscoveredbyPerelman[17].Forexample,consideringgeodesicdistanceinoursolitonmet

4、ricsgivesrisetoPerelman'sL-length.ItisalsofruitfultoconsiderexistingRicci owtheoryappliedtotheCanonicalSoliton ows.Forexample,applyingtheoryofMcCannandthesecondauthorandIlmanen[15]isonewayofleadingtotheresultsof[19]whichultimatelyrecoversessentiallyallofthemonotonicquantitiesfo

5、rRicci owusedbyPerelman[17].See[20]forabroaderdescription.InthispaperwedescribeaslightvariationoftheCanonicalShrinkingSolitons{namelytheCanonicalExpandingSolitons{whichhavecompletelydierentapplications.ThesenewsolitonsareadaptedtoexplainingandprovingHarnackinequalitiesinthespi

6、ritoftheoriginalresultofHamilton[10]andthemorerecentresultofBrendle[2].OurworkrecoversbothoftheseknownHarnackinequalities,andgivesnewonestoo.(SeeTheorem2.7.)Inaddition,ourmethodexplainsclearlywhatisbehindaHarnackinequality:itissimplytheassertionthatagivencurvatureconditionispre

7、servedontheCanonicalExpandingSoliton.Asaby-productofourworkwegiveananswertothequestionofWallachandHamilton[11]whichasksforageometricconstructionwhosecurvatureisrepresentedbythematrixHarnackquantityofHamilton[10].ThisquestionpromptedthepioneeringworksofChow-Chu[5](seealsotherele

8、vantmodicationin[7,chapter11,x1.3])andChow-Knopf[6]wh