Ricci flow, entropy and optimal transportation

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1、RICCIFLOW,ENTROPYANDOPTIMALTRANSPORTATIONROBERTJ.MCCANNyANDPETERM.TOPPINGzAbstract.LetasmoothfamilyofRiemannianmetricsg()satisfythebackwardsRicci owequationonacompactorientedn-dimensionalmani-foldM.Supposetwofamiliesofnormalizedn-forms!()0and~!()0satisfytheforwards(in)h

2、eatequationonMgeneratedbytheconnectionLaplacian
g().Ifthesen-formsrepresenttwoevolvingdistributionsofparticlesoverM,theminimumroot-mean-squaredistanceW2(!();!~();)totransporttheparticlesof!()ontothoseof~!()isshowntobenon-increasingasafunctionof,withoutsignconditionsonth

3、ecurvatureof(M;g()).Moreover,thiscontractivitypropertyisshowntocharacterisesupersolutionstotheRicci ow.1.introductionOnacompactorientedn-dimensionalmanifoldM,letg()beasmoothfamilyofmetricsfor2[1;2].Weareparticularlyinterestedinthecasethatg()satisesthebackwardsRicci oweq

4、uation@g(1)=2Ric(g)@whereRic(g)istheRiccitensorofg.Giventerminaldatag(2),suchafamilycanalwaysbeconstructedfor1sucientlycloseto2(seeHamilton[11],DeTurck[9],[27,Ch.5]).Thegeodesicdistanced(x;y;)betweentwopointsx;y2M,withrespecttog(),evolvesaccordingtotheformulaZ12(2)d2(x;

5、y;)=infdds;(0)=x;(1)=y0dsg()wheretheinmumistakenoversmoothcurves:[0;1]!Mjoiningxtoy.Similarly,giventwoBorelprobabilitymeasuresand~onM,the2-WassersteindistanceW2(;;~)betweenthemevolvesaccordingtoitsde-nitionZ22(3)W2(;;~)=infd(x;y;)d(x;y):2(;~)MMDate:March

6、14,2008.ThispapersupercedestheearlierpaperDiusionisa2-WassersteincontractiononanymanifoldevolvingbyreverseRicci ow(2006).c2007,theauthors.12Theinmumistakenoverthespace(;~)ofBorelprobabilitymeasuresonMMwhichhavemarginalsand~,inthesensethatZZf(x)d(x)=f(x)d(x;y);andMM

7、M(4)ZZf(y)d(x;y)=f(y)d~(y);MMMforeachcontinuoustestfunctionf2C(M).Inthispaper,weareparticularlyinterestedinthecaseofmeasuresand~whichareinducedbyn-forms!and~!respectively,inthesensethatZ(A)=!;AforeveryBorelAM,andsimilarlyfor~!.(Wewilloftencorruptnotationbyconsideringth

8、eWassersteindistancebetween!and~!ratherthanand~.)Theadvanta