Global Pinching Theorems of Submanifolds in Spheres

Let M be a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphere Sn+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 Sn+1.