RIGIDITY OF MINIMAL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE

Abstract

Abstract. Let M n be a complete immersed super stable minimal sub-manifold in R n + p with °at normal bundle. We prove that if M has flnitetotal L 2 norm of its second fundamental form, then M is an a–ne n -plane. We also prove that any complete immersed super stable minimalsubmanifold with °at normal bundle has only one end. 1. IntroductionLet M be an n -dimensional complete minimal submanifold in R n + p . When n = 2 and p = 1, do Carmo and Peng [3], Fischer-Colbrie and Schoen [5]independently showed that the only complete stable minimal surface is a plane.Recall that a minimal submanifold is stable if the second variation of its volumeis always nonnegative for any normal variation with compact support. Forhigher dimensional minimal hypersurfaces, do Carmo and Peng [4] generalizedthe result mentioned as above. We will denote by A the second fundamentalform of M .Theorem ([4]). Let M n be a complete stable minimal hypersurface in R n +1 satisfying R M jAj 2 dv < 1. Then M must be a hyperplane.