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 S n + p ( n ≥ 2 , p ≥ 1 ) . By using the Sobolev inequalities of P. Li (1980) to L p 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 L p 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 .