WILLMORE SURFACES IN THE UNIT N-SPHERE

Abstract

Let $M^2$ be a compact Willmore surface in the $n$-dimensional unit sphere. Denote by $\phi_{ij}^{\alpha}$ the tracefree part of the second fundamental form $h_{ij}^{\alpha}$ of $M^2,$ and by $\mathbb{H}$ the mean curvature vector of $M^2.$ Let $\Phi$ be the square of the length of $\phi_{ij}^{\alpha}$ and $H=|\mathbb{H}|$. We prove that if $0\leq\Phi\leq C(1+\frac{H^2}{8})$, where $C=2$ when $n=3$ and $C=\frac{4}{3}$ when $n \geq 4$, then either $\Phi=0$ and $M^2$ is totally umbilic or $\Phi=C(1+\frac{H^2}{8})$. In the latter case, either $n=3$ and $M^2$ is the Clifford torus or $n=4$ and $M^2$ is the Veronese surface.