Spherical rigidities of submanifolds in Euclidean spaces

Abstract

In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space En+p. We prove that if Mn is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in En+p and satisfies either: (1)S≤n2H2n-1, or (2)n2H2≤(n-1)Rn-2, then Mn is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of Mn and the scalar curvature of Mn, respectively. On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere Sn(c), the totally geodesic Euclidean space En, and the generalized cylinder Sn-1(c)×E1 are only n-dimensional (n>2) complete connected submanifolds Mn with constant mean curvature H in En+p if S≤n2H2/(n-1) holds.