A global pinching theorem for surfaces with constant mean curvature in S-3

Abstract

Let M be a compact immersed surface in the unit sphere S-3 with constant mean curvature H. Denote by phi the linear map from T-p(M) into Tp(M), phi = A - H/2 I, where A is the linear map associated to the second fundamental form and I is the identity map. Let Phi denote the square of the length of phi. We prove that if parallel to Phi parallel to (L2) less than or equal to C, then M is either totally umbilical or an H(r)-torus, where C is a constant depending only on the mean curvature H.