Area minimizing vector fields on round spheres Vincent Borrelli and Olga Gil Medrano

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Niveau: Supérieur, Doctorat, Bac+8
Area minimizing vector fields on round 2-spheres Vincent Borrelli and Olga Gil-Medrano Abstract. – A vector field V on a n-dimensional round sphere Sn(r) defines a submanifold V (Sn) of the tangent bundle TSn. The Gluck and Ziller question is to find the infimum of the n-dimensional volume of V (Sn) among unit vector fields. This volume is computed with respect to the natural metric on the tangent bundle as defined by Sasaki. Surprisingly, the problem is only solved for dimension three [10]. In this article we tackle the question for the 2-sphere. Since there is no glob- ally defined vector field on S2, the infimum is taken on singular unit vector fields without boundary. These are vector fields defined on a dense open set and such that the closure of their image is a surface without boundary. In particular if the vector field is area minimizing it defines a minimal surface of T 1S2(r). We prove that if this minimal surface is homeomorphic to RP 2 then it must be the Pontrya- gin cycle. It is the closure of unit vector fields with one singularity obtained by parallel translating a given vector along any great circle passing through a given point. We show that Pontryagin fields of the unit 2-sphere are area minimizing. 2000 Mathematics Subject Classification : 53C20 Keywords and phrases : Minimal submanifold, vector field, area minimiz- ing surface, Pontryagin cycles.

  • vector field

  • riemannian manifold

  • ing surface

  • dimensional spheres

  • any great

  • field should

  • projective space

  • tryagin cycles

  • metric


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Areaminimizingvectorfieldsonround2-spheresVincentBorrelliandOlgaGil-MedranoAbstract.–AvectorfieldVonan-dimensionalroundsphereSn(r)definesasubmanifoldV(Sn)ofthetangentbundleTSn.TheGluckandZillerquestionistofindtheinfimumofthen-dimensionalvolumeofV(Sn)amongunitvectorfields.ThisvolumeiscomputedwithrespecttothenaturalmetriconthetangentbundleasdefinedbySasaki.Surprisingly,theproblemisonlysolvedfordimensionthree[10].Inthisarticlewetacklethequestionforthe2-sphere.Sincethereisnoglob-allydefinedvectorfieldonS2,theinfimumistakenonsingularunitvectorfieldswithoutboundary.Thesearevectorfieldsdefinedonadenseopensetandsuchthattheclosureoftheirimageisasurfacewithoutboundary.InparticularifthevectorfieldisareaminimizingitdefinesaminimalsurfaceofT1S2(r).WeprovethatifthisminimalsurfaceishomeomorphictoRP2thenitmustbethePontrya-gincycle.Itistheclosureofunitvectorfieldswithonesingularityobtainedbyparalleltranslatingagivenvectoralonganygreatcirclepassingthroughagivenpoint.WeshowthatPontryaginfieldsoftheunit2-sphereareareaminimizing.2000MathematicsSubjectClassification:53C20Keywordsandphrases:Minimalsubmanifold,vectorfield,areaminimiz-ingsurface,Pontryagincycles.1IntroductionandmainresultsThevolumeVol(V)ofasmoothunitvectorfieldV:MnT1Mnisthen-dimensionalvolumeofitsimageV(M)asasubmanifoldoftheunittangentbundleT1M.Anaturalquestion,thatgoesbacktothepioneeringworkofH.GluckandW.Ziller[10],istoaskforthe(absolute)minimizersofthevolume.Moreprecisely,givenanorientedcompactRiemannianmanifold(M,g)thequestionistodetermineVΓin(fT1M)Vol(V)1