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

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