International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 3, Pages 183-191
doi:10.1155/S0161171202106247

Global pinching theorems of submanifolds in spheres

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.