Kairen Cai

Copyright © 2002 Kairen Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let M be a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphere S n+p(n≥2 ,p≥1). By using the Sobolev inequalities of P. Li (1980) to Lp estimate for the square length σ of the second fundamental form and the norm of a tensor Φ, related to the second fundamental form, we set up some rigidity theorems. Denote by ‖σ‖p the Lp norm of σ and H the constant mean curvature of M. It is shown that there is a constant C depending only on n, H, and k where (n−1) k is the lower bound of Ricci curvature such that if ‖σ‖ n/2<C, then M is a totally umbilic hypersurface in the sphere S n+1.