on the volume functional of compact manifolds with boundary with constant scalar curvature

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1、ONTHEVOLUMEFUNCTIONALOFCOMPACTMANIFOLDSWITHBOUNDARYWITHCONSTANTSCALARCURVATUREPENGZIMIAOANDLUEN-FAITAM1Abstract.Westudythevolumefunctionalonthespaceofcon-stantscalarcurvaturemetricswithaprescribedboundarymetric.Wederiveasufficientandnecessaryconditionforametr

2、ictobeacriticalpoint,andshowthattheonlydomainsinspaceforms,onwhichthestandardmetricsarecriticalpoints,aregeodesicballs.Inthezeroscalarcurvaturecase,assumingtheboundarycanbeiso-metricallyembeddedintheEuclideanspaceasacompactstrictlyconvexhypersurface,weshowt

3、hatthevolumeofacriticalpointisalwaysnolessthantheEuclideanvolumeboundedbytheisomet-ricembeddingoftheboundary,andthetwovolumesareequalifandonlyifthecriticalpointisisometrictoastandardEuclideanball.Wealsoderiveasecondvariationformulaandapplyittoshowthat,onEuc

4、lideanballsand“small”hyperbolicandspheri-calballsindimensions3≤n≤5,thestandardspaceformmetricsareindeedsaddlepointsforthevolumefunctional.1.IntroductionGivenacompactn-dimensionalmanifoldΩwithaboundaryΣ,westudyvariationalpropertiesofthevolumefunctionalonthes

5、paceofconstantscalarcurvaturemetricsonΩwithaprescribedboundarymetriconΣ.Thedimensionnisassumedtobe≥3.Thereareseveralmotivationsforustoconsiderthisproblem.arXiv:0807.2693v1[math.DG]17Jul2008Thefirstmotivationcomesfromarecentresultin[9].Thereoneconsidersanasym

6、ptoticallyflat3-manifold(M,g)withagivenend.Let{xi}beacoordinatesystemat∞whichdefinestheasymptoticstructureof(M,g).LetSr={x∈M

7、

8、x

9、=r}bethecoordinatesphere,where

10、x

11、denotesthecoordinatelength.LetγbetheinducedmetriconSr.Whenrislarge,(Sr,γ)canbeisometricallyembedde

12、dintheEuclideanspaceR3asastrictlyconvexhypersurfaceS0.LetV(r)r0Date:June2008.2000MathematicsSubjectClassification.Primary53C20;Secondary58JXX.1ResearchpartiallysupportedbyEarmarkedGrantofHongKong#CUHK403005.12PengziMiaoandLuen-FaiTambethevolumeoftheregionenc

13、losedbyS0inR3andVbethevolumerroftheregionenclosedbySin(M3,g).Itwasprovedin[9]that,asrr→∞,22(1)V(r)−V0(r)=2mADMπr+o(r)whenevermisdefined.HeremistheADMmassof(M3,g)ADMADM[2].ThereforeifR(g),thescalarcurvat