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 $T_p(M)$, $\phi =A-\frac H2I$, 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 $||\Phi ||_{L^2}\leq C$, then $M$ is either totally umbilical or an $H(r)$-torus, where $C$ is a constant depending only on the mean curvature $H$.