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1、复Finsler流形上的Hodge-Laplace算子第35卷第4期2006年8月数学进展ADVANCESIN,IATHEIATICS}1.35No4Aug.,2006Hedge--LaplaceOperatoronComplexFinslerManifoldsZHONGChun—ping,.ZHONGTong—def1.DepartmentelMathematics,Zh~iangUni,~ersity,Hangzhou.Zh~iang,310028,P.R.China;2.SchoolelMathematicalScience,XiamenUniversity,Xiamen
2、,Fujian,361005,P..China)Abstract:AHedge—Iaplaceoperatorisdeftnedonacompactstrongl3'pseudoconxexcomplexFinslerman[fold(M,F1,itreducestotheclassicalHodge-Iaplaveoperatorin1IermitiancasesThekeypointofdefiningthisHedge—Iaplaceoperatoristodefineaglobalinnerproductonthebasemanifolcl^,.wdothisbypul
3、lingthedifferentialformsoftype一(p,q)O11AIbacktotheprojectixizedIangentbundle1PT^of^,,andthenusingthenaturalHermitianinnerproductonPT^,toobtainagloba1innerproductonM.Keywords:complexFinlsermanifold;Hedge—Iaplaceoperator;projectiizedtmigentbmldleMR(1991)SubjectClassiflcation:32C10;32c17;53C60
4、/CLCnumber:OI86.14Documentcode:AArtieleID:1000—0917(2006104—0415—121IntroductionitiswellknownthattheLaplacianplaysanimportantroleinthetheoryofharmonicintegralandBochnertechniquebothinRiemannianandKRidermanifolds.ForRiemannLaplacianOllRiemannianmanifoldsand0LaplacianonHermitianmanifolds,weref
5、ertor1.2I.LaplacianalsomakesenseinthemoregeneralFinslerspaces.RecentlyundertheinitiationofS.S.CherntheglobaldifferentialgeometryofrealandcomplexFinslmmanifoldshasgainedagreatdevelopment[3—61.D.BaoandB.Lackey[7]havesuccessfullydefinedaHedge—LaplaceoperatoronacompactrealFinslermanifoldandobtai
6、nedaHedgedecompositiontheorem,whichmaketheBochnertypevanishingtheoremsviable.ThereareotherLaplaciansandtheirapplicationsontealFinslermanifolds[8一引.ButuptonowtherearenoresultsforLaplaciananditsapplicationsoncomplexFinslermanifolds.Asweknow.thetealFinslermetricassociatedtoacomplexFinslermetric
7、isnot,necessarilystronglyconvex.evenifthecomplexFinslermetricisstronglyconvex.ThustherealFinslermetricandthecomplexFinslermetricarelooselyrelated,whichdifie,rsfromtherelationshipbetweenRiemannianmetricandHermitianmetric.SothattheconsiderationofLapl
