What is wrong with the Hausdorff measure in Finsle

arXiv:math/0408413v1[math.DG]30Aug2004WHATISWRONGWITHTHEHAUSDORFFMEASUREINFINSLERSPACESJ.C.´ALVAREZPAIVAANDG.BERCKAbstract.WeconstructaclassofFinslermetricsinthree-dimensionalspacesuchthatalltheirgeodesicsarelines,butnotallplanesareex-tremalfortheirHausdorffareafunctionals.ThisshowsthatiftheHaus-dorffmeasureisusedasnotionofvolumeonFinslerspaces,thentotallygeodesicsubmanifoldsarenotnecessarilyminimal,fillingresultssuchasthoseofIvanov[18]donothold,andintegral-geometricformulasdonotexist.Ontheotherhand,usingtheHolmes-Thompsondefinitionofvolume,weproveageneralCroftonformulaforFinslerspacesandgiveaneasyproofthattheirtotallygeodesichypersurfacesareminimal.1.IntroductionOneofthefirstquestionsthatariseinthestudyofFinslermanifoldsiswhetheritispossibletodefinetheirvolumeinanaturalway.In[12]BusemannarguesstronglythatthevolumeofaFinslermanifoldshouldbeitsHausdorffmeasure.HisargumentisbasedonanumberofaxiomsthatanynaturaldefinitionofvolumeonFinslerspacesmustsatisfy.Alas,thereareothernaturaldefinitionsofvolumethatsatisfythoseaxioms.AmongthemisonethatisrapidlybecomingtheuncontesteddefinitionofvolumeforFinslerspaces:theHolmes-Thompsonvolume.Definition1.1.TheHolmes-Thompsonvolumeofann-dimensionalFinslermanifoldsequalsthesymplecticvolumeofitsunitco-discbundledividedbythevolumeoftheEuclideanunitballofdimensionn.TheareaofasubmanifoldistheHolmes-ThompsonvolumefortheinducedFinslermetric.Itsdeeptiestoconvexity,differentialgeometry,andintegralgeometry(see[21],[5],and[1])havemadetheHolmes-ThompsonvolumeeasiertostudythantheHausdorffmeasure.However,sincemanyofthetheoremsthatareknowntoholdfortheHolmes-ThompsonvolumehavenotbeenprovednordisprovedfortheHausdorffmeasure,onemaywonderwhetheritisjustnotsuitableforthestudyofFinslerspaces,orwhethertherighttechniquesforitsstudyhavenotyetbeenfound.OurresultssuggestthattheHausdorffmeasureisnotsuitable.1991MathematicsSubjectClassification.53B40;49Q05,53C55.Keywordsandphrases.Finslermanifold,Hausdorffmeasure,minimalsubmanifolds,integralgeometry,Holmes-Thompsonvolume.ThefirstauthorwaspartiallyfundedbyFAPESPgrantNo2004/01509-0.12J.C.´ALVAREZPAIVAANDG.BERCKInthispaperweshowthroughanexperimentthattotallygeodesicsub-manifoldsofaFinslerspacedonotnecessarilyextremizetheHausdorffareaintegrand.TheexperimentalsoshowsthattheHolmes-ThompsonvolumecannotbereplacedbytheHausdorffmeasureinthefillingresultofS.Ivanov[18],andthatthereisnoCroftonformulafortheHausdorffmeasureofhy-persurfacesinFinslerspaces.In[20]SchneidergivesexamplesofnormedspacesforwhichthereisnoCroftonformulafortheHausdorffmeasureofhypersurfaces,buthisexamplesarenotFinslerspaces—theirunitspheresarenotsmoothhypersurfaceswithpositiveGaussiancurvature.Thephe-nomenonweuncoveriscompletelydifferent:theobstructiontotheexistenceofaCroftonformuladoesnotlieinthenormedspacesthatmakeupthetangentbundle,butinthewaytheycombinetomakeaFinslermetric.Infact,wecanfindFinslermetricsinR3whoserestrictiontoalargeballisarbitrarilyclosetotheEuclideanmetricandforwhichthereisnoCroftonformulafortheHausdorffmeasureofhypersurfacescontainedintheball.UsingthestandardnormandinnerproductinR3andidentifyingR3×R3withthetangentbundleofR3,theexperimentcanbesummarizedinthefollowingtheorem.Theorem.Foranyrealnumberλ,allgeodesicsoftheFinslermetricϕλ(x,v)=(1+λ2kxk2)kvk2+λ2hx,vi2kvkarestraightlines.However,theonlyvalueofλforwhichallplanesareextremalsoftheHausdorffareafunctionalofϕλisλ=0.Thereisalotmoreinthistheoremthatmeetstheeye.Forexample,ittakessomeworkinSection2justtoseethattheϕλareFinslermetrics.ThecomputationofareaintegrandsforFinslermetricsinR3requiresthecomputationofaFunktransform.ItisonlyduetothesimpleformulaforϕλthatinSection3weareabletocomputeitsareaintegrandinR3×Λ2R3:φλ(x,a)=2(1+λ2kxk2)3/2(1+2λ2kxk2)kak2−λ2hx,ai23/2(2+3λ2kxk2)kak2−λ2hx,ai2.Apparently,thisisthefirstexampleofanexplicitcomputationofthe(Haus-dorff)areaintegrandofanon-RiemannianFinslermetric.InSection4wecharacterizesmoothparametricintegrandsofdegreen−1onRnforwhichhyperplanesareextremalasthosethatsatisfyacertainlineardifferentialequation.ThesimplicityofthisequationmakesitpossibletoshowthatallplanescannotbeextremalsfortheHausdorffareafunctionalofϕλunlessλiszero.AlsoinSection4wequicklyshowthatIvanov’sfillingtheoremimpliesthattwo-dimensionaltotallygeodesicsubmanifoldsofFinslerspacesareminimalfortheHolmes-Thompsonareafunctional,andconcludethatfillingtheoremsofthisnaturecannotholdfortheHausdorffmeasure.WHATISWRONGWITHTHEHAUSDORFFMEASUREINFINSLERSPACES3TheproofoftheCroftonformulaforhypersurfacesinFinslerspaces,itsapplicationtotheminimalityoftotallygeodesichypersurfacesfortheHolmes-Thompsonvolume,andtheproofthatthereisnoCroftonformulafortheHausdorffareafunctionalofϕλunlessλiszeroareallinSection52.AclassofFinslermetricsRoughlyspeaking,aFinslermanifoldisamanifoldinwhicheachtangentspacehasbeenprovidedwithanormandthesenormsvarysmoothlywiththebasepoint.Inordertoguaranteethatthesubjectstaysintherealmofdifferentialgeometry,itisstandard(see[7])toaskthatthenormsbeMinkowskinorms.Definition2.1.LetVbeafinite-dimensionalvectorspaceoverthereals.AnormϕonVissaidtobeaMinkowskinormif