Some discretizations of geometric evolution equati

arXiv:0709.0990v1[math.DG]7Sep2007SomediscretizationsofgeometricevolutionequationsandtheRicciiterationonthespaceofK¨ahlermetrics,IYanirA.Rubinstein∗Abstract.Inthisarticleandinitssequelweproposethestudyofcertaindiscretizationsofgeometricevolutionequationsasanapproachtothestudyoftheexistenceproblemofsomeellipticpartialdifferentialequationsofage-ometricnatureaswellasameanstoobtaininterestingdynamicsoncertaininfinite-dimensionalspaces.WeillustratethefruitfulnessofthisapproachinthecontextoftheRicciflow,aswellasanotherflow,inK¨ahlergeometry.WeintroduceandstudydynamicalsystemsrelatedtotheRiccioperatoronthespaceofK¨ahlermetricsthatariseasdiscretizationsoftheseflows.Weposesomeproblemsregardingtheirdynamics.Wepointoutanumberofap-plicationstowell-studiedobjectsinK¨ahlerandconformalgeometrysuchasconstantscalarcurvaturemetrics,K¨ahler-Riccisolitons,Nadel-typemultiplieridealsheaves,balancedmetrics,theMoser-Trudinger-Onofriinequality,energyfunctionalsandthegeometryandstructureofthespaceofK¨ahlermetrics.E.g.,weobtainanewsharpinequalitystrengtheningtheclassicalMoser-Trudinger-Onofriinequalityonthetwo-sphere.Contents1.Introduction............................................................22.ConstructingcanonicalmetricsinK¨ahlergeometry......................43.TheRicciiteration......................................................64.SomeenergyfunctionalsonthespaceofK¨ahlermetrics................105.TheRicciiterationfornegativeandzerofirstChernclass..............146.TheRicciiterationforpositivefirstChernclass........................157.TheK¨ahler-RicciflowandtheRicciiterationforageneralK¨ahlerclass.188.AnotherflowandtheinverseRiccioperatorforageneralK¨ahlerclass..199.ThetwistedRicciiterationandatwistedinverseRiccioperator........22∗MassachusettsInstituteofTechnology.Email:[e-mail]6th,2007.MathematicsSubjectClassification(2000):Primary32W20.Secondary14J45,26D15,32M25,32Q20,39A12,53C25,58E11.1Y.A.Rubinstein10.Someapplications.....................................................2410.1TheMoser-Trudinger-OnofriinequalityontheRiemannsphereanditshigher-dimensionalanalogues................................2410.2AnanalyticcharacterizationofK¨ahler-Einsteinmanifoldsandananalyticcriterionforalmost-K¨ahler-Einsteinmanifolds...............2710.3AnewMoser-Trudinger-OnofriinequalityontheRiemannsphereandafamilyofenergyfunctionals...................................2910.4ConstructionofNadel-typeobstructionsheaves......................3110.5Relationtobalancedmetrics........................................3310.6AquestionofNadel.................................................3310.7TheRicciindexandacanonicalnestedstructureonthespaceofK¨ahlermetrics......................................................35Bibliography.............................................................361Introduction.Ourmainpurposeinthisarticleandinitssequel[R5]istoproposethesystematicuseofcertaindiscretizationsofgeometricevolutionequationsasanapproachtothestudyoftheexistenceproblemofcertainellipticpartialdifferentialequationsofageometricnatureaswellasameanstoobtaininterestingdynamicsoncertaininfinite-dimensionalspaces.WeillustratethefruitfulnessofthisapproachinthecontextoftheRicciflow,aswellasanotherflow,inK¨ahlergeometry.WedescribehowthisapproachgivesanewmethodfortheconstructionofcanonicalK¨ahlermetrics.WealsointroduceanumberofcanonicaldynamicalsystemsonthespaceofK¨ahlermetricsthatwebelievemeritfurtherstudy.Someoftheresultsandconstructionsdescribedherewereannouncedpreviously[R3].Givenanellipticpartialdifferentialequation,severalclassicalmethodsareavail-abletoapproachtheproblemofexistenceofsolutions.Inessence,standardelliptictheoryreducestheexistenceproblemtothedemonstrationofcertainaprioriesti-matesforsolutions.Themaindifficultyliesthereforeindevisingmethodstoobtaintheseestimates.Onecommonmethod,thatgoesbackatleasttoBernsteinandPoincar´e,isthecon-tinuitymethod.Inthisapproachonecontinuouslydeformsthegivenellipticoperatortoanother(oftentimesinalinearfashion),forwhichtheexistenceproblemisknowntohavesolutions.Ellipticityprovidesforexistenceofsolutionsforsmallperturba-tionsofthiseasierproblem.Inordertoproveexistenceforthewholedeformationpathonethenseekstoestablishaprioriestimates,uniformalongthedeformation,forsolutionsofthefamilyofellipticproblems.Anotherapproach,drawingsomeofitsmotivationfromPhysics,istheheatflowmethod,goingbacktoFourier.Heretheideaisstudyadeformationoftheellipticproblemaccordingtoaparabolicheatequationwhoseequilibriumstateispreciselyasolutiontotheoriginalellipticequation.Muchofthestandardelliptictheoryhasa2Somediscretizationsofgeometricevolutionequationsparaboliccounterpart.First,onemakesuseofthelatterinordertoestablishshort-timeexistence.Long-timeexistenceandconvergencethenhingeuponestablishingaprioriestimates,asbefore.Athirdapproach,goingback,amongothers,toEulerandCauchy,isthediscretiza-tionmethod,thatcanbeconsideredasablendofthetwoabove.Heretheideaistoreplaceanevolutionequation(or“flow”)byacountablesetofellipticequationsthatarisebyrepeatedlysolvingadiffer