Some rigidity theorems for minimal submanifolds of the sphere

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1、SOMERIGIDITYTHEOREMSFORMINIMALSUBMANIFOLDSOFTHESPHEREBYD.FISCHER-COLBRIE(1)ColumbiaUniversity,NewYork,N.Y.,U.S.A.IntroductionItiswellknownthattheregularityofminimalsubmanifoldscanbereducedtothestudyofminimalconesandhencetocompactminimalsubmanifoldsofthesphere.Aphe-nomenonrelatedtoregularitywasdis

2、coveredbyBernstein[3]in1915.TheBernsteinTheoremsaysthatanentiresolutiontotheminimalsurfaceequationintwovariablesisaplane.Ananswertotheextrinsicrigidityquestionofwhetheraminimalsubmanifoldwhichliesinsomeneighborhoodofastandardspheremustactuallybeastandardsphereisofinterestinrelationtotheabovetopic

3、saswellasinitsownright.EffortstogeneralizeBernstein'sTheoremweremadebymanyauthors.TheworkofSimons[16]completedtheproofincodimensiononeuptodimension7.Bombieri,deGiorgiandGuisti[4]gaveacounterexampleindimension8.Fortwo-dimensionalgraphs,Osserman[14]provedaversionofBcrnstein'stheoremassumingthenorma

4、lvectorsomitaneighborhoodofthesphere.Simons[16]provedthataminimalconewhosenormalplaneslieinasufficientlysmallneighborhoodisaplaneandhenceBernstein'stheoremistrueforagraphwhosenormalssatisfythesamecondition.Reilly[15]enlargedtheneighbor-hood.Hisestimatesaysthatifaconehasthepropertythatthenormalssa

5、tisfy>V(2k-2)/(3k-2)forsomefixedk-planeA,thentheconeisaplane.Fortwo-dimensionalminimalgraphsBarbosa[2]improvedtheneighborhoodstoanopenhemisphere.Morespecifically,BarbosashowedthatacompactminimallyimmersedsphereinS~+~suchthatitsnormalsatisfy>0forsomefixedAistotallygeodesic.Thetheoremwasa

6、lsoprovedbyS.T.Yau[17]forS2inS4andbyKenmatsu[10]underthestrongerassump-tionofaboundon.LawsonandOsserman[12]constructedaseriesofexamplesof(1)SupportedinpartbyNSFgrantMGS77-18723A01.30D.FISCHER-COLBRIEminimalgraphswhichareconesregularawayfromtheorigin,thusshowingthatBarbosa'sTheoremfailstohold

7、ingeneraldimensionsandcodimensionsandregularitydoesnotholdforLipschitzsolutionstotheminimalsurfacesystem.Thispaperdealswiththestudyofminimalconeswhosenormalssatisfyconditionsofthistype.WeimprovetheestimatesofReillyandw