The canonical shrinking soliton associated to a Ricci f

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1、thecanonicalshrinkingsolitonassociatedtoaricciflowEstherCabezas-RivasandPeterM.Topping4February2010AbstractToeveryRicci owonamanifoldMoveratimeintervalIR,weassociateashrinkingRiccisolitononthespace-timeMI.WerelatepropertiesoftheoriginalRicci owtopropertiesofth

2、enewhigher-dimensionalRicci owequippedwithitsowntime-parameter.Thisgeometricconstructionwasdiscoveredbyconsiderationofthetheoryofoptimaltransportation,andinparticulartheresultsofthesecondauthor[18],andMcCannandthesecondauthor[12];webrie ysurveythelinkbetweenthese

3、subjects.1IntroductionIn1982,Hamilton[7]introducedthestudyofRicci ow,whichevolvesaRiemannianmetricgonamanifoldMunderthenonlinearevolutionequation@g=2Ric(g(t));(1.1)@tfortinsometimeintervalIR.Sincethen,thesubjecthasdevelopedsteadily,andhasbecomeestablishedasane

4、ectivebridgebetweenanalysis,geometryandtopology(seeforexample[13],[14],[15]andtheoverviewin[17]).TheinitialprogressrelevanttothepresentpaperwasHamilton'sdiscoveryin1993oftheso-calledHarnackquantities(see[8]formoreinformation)andby1995,NolanWallach[9,x14]hadpropos

5、edthatthesequantitiesshouldariseassomesortofcurvatureofsomehigher-dimensionalmanifoldorbundleassociatedtotheRicci ow.ThisideawasdevelopedbyChowandChu[2]whoconsideredthespace-timemanifoldMI,anddenedapair(~g;re)ofametriconitscotangentbundledegenerateinthetimedire

6、ctionanda~g-compatibletorsion-freeconnection(whichisnotuniqueowingtothedegeneracyof~g)sothatthederivativesinthetimecoordinatedirectionofthecomponentsof(~g;re)resembletheformulaeonecancomputefortheevolutionofthecomponentsofthemetricanditsLevi-Civitaconnectionunder

7、Ricci ow.(See[2]formoredetails.)ItturnsoutthatHamilton'smatrixHarnackquadraticisalmosttheRiemanniancurvatureofthatspace-timeconnection.AnimprovedcorrespondenceisestablishedintheworkofChowandKnopf[4]byconsideringRicci owwitha`cosmologicalterm'.(Anexampleofsucha ow

8、wouldbeg(t):=1g(t),fort=logt,whereg(t)isaRicci ow.)tIn2002,Perelman[13,x6]madeanewbreakthroughalongtheselinesinvolvingtheconstructionofanessentiallyRicci- a