L_n/2-pinching theorem for submanifolds in a sphere

Abstract

Let Mⁿ (n ≥ 2) be a n-dimensional oriented closed submanifolds with parallel mean curvature in S^{n + p} (1), denote by S, the norm square of the second fundamental form of M. H is the constant mean curvature of M. We prove that if ∫_M S^n/2 ≤ A (n), where A (n) is a positive universal constant, then M must be a totally umbilical hypersurface in the sphere S^{n + 1}.