Global Pinching Theorem for Hypersurfaces with Constant Mean Curvature in Sphere

Let M be a compact embedded hypersurface with constant mean curvature H and positive Ricci curvature in the unit sphere Sn+1. By using the Sobolev inequalties of P.Li to the norm of a tensor φ, related to the second fundamental form, we set up a pinching theorem. Denote by ||φ||p the Lv norm of the square length of the second fundamental form. 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 totally umbilic.