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1、ISSN1464-89971Geometry&TopologyMonographsVolume1:TheEpsteinBirthdaySchriftPages1{21ThemeancurvatureintegralisinvariantunderbendingFredericJAlmgrenJrIgorRivinAbstractSupposeMtisasmoothfamilyofcompactconnectedtwodi-3mensionalsubmanifoldsofEuclideanspaceEwi
2、thoutboundaryvaryingisometricallyintheirinducedRiemannianmetrics.ThenweshowthatthemeancurvatureintegralsZ2HtdHMtareconstant.ItisunknownwhethertherearenontrivialsuchbendingsMt.Theestimatesalsoholdforperiodicmanifoldsforwhichtherearenontrivialbendings.Inad
3、dition,ourmethodsworkessentiallywithoutnnchangetoshowthesimilarresultsforsubmanifoldsofHandS,towit,ifMt=@XtZ2dHtdH=?kn?1dV(Xt);Mt33wherek=?1forHandk=1forS.TheEuclideancasecanbeviewedasaspecialcasewherek=0.Therigidityofthemeancurvatureintegralcanbeusedtos
4、hownewrigidityresultsforisometricembeddingsandprovidenewproofsofsomewell-knownresults.This,togetherwithfar-reachingextensionsoftheresultsofthepresentnoteisdoneinthepreprint[6].Ourresultshouldbecomparedwiththewell-knownformulaofHerglotz(see[5],also[8]and[
5、2]).AMSClassi cation53A07,49Q15KeywordsIsometricembedding,integralmeancurvature,bending,varifolds1IntroductionTheunderlyingideaofthisnoteisthefollowing.SupposeNtisasmoothlyvary-ingfamilyofpolyhedralsolidshavingedgesEt(k),andassociated(signed)kdihedrala
6、nglest(k).AccordingtoatheoremofSchla?i[7]kXdEt(k)t(k)=0:dtkCopyrightGeometryandTopology2FredericJAlmgrenJrandIgorRivinIncaseedgelengthispreservedinthefamily,iedEt(k)=0dtforeachtimetandeachk,thenalso(productrule)dXEt(k)t(k)=0:dtkShouldthe@Nt'sbepolyhed
7、ralapproximationstosubmanifoldsMtvaryingisometrically,onemightregardXEt(k)t(k)kasareasonableapproximationtothemeancurvatureintegralsZHdH2tMtandexpectdEt(k)dttobesmall.HenceitisplausiblethatthemeancurvatureintegralsoftheMt'smightbeconstant.Inthisnotewesh
8、owthatthatisindeedthecase.Examplessuchastheisometrypicturedonpage306ofvolume5of[8]showthatthemeancurvatureintegralisnotpreservedunderdiscreteisometries.Twocommentsareinorder.The rstisthatitisverylikelythattherearenoisometr
