WILLMORE SURFACES AND F-WILLMORE SURFACES IN SPACE FORMS

Abstract

Let $M^{2}$ be a compact $F$-Willmore surface in the $n$-dimensional space form $\mathbb{N}^{n}(c)$ of constant curvature $c$. Denote by $\phi_{ij}^{\alpha}$ the trace free part of the second fundamental form $h=(h_{ij}^{\alpha})$, and by $\mathbf{H}$ the mean curvature vector of $M^{2}$. Let $\Phi$ be the square of the length of $\phi_{ij}^{\alpha}$ and $H = |\mathbf{H}|$. If $F^{\prime}(\Phi)\geq 0$, then $\int_{M} \Big\{F^{\prime\prime}(\Phi)\Big[\frac{1}{2}|\nabla\Phi|^{2} - \sum_{\alpha,i,j} \phi_{ij}^{\alpha}\Phi_{j}H_{i}^{\alpha}\Big]+F(\Phi)H^{2} + F^{\prime}(\Phi) (2c-K(n)\Phi)\Phi\Big\} dv \leq 0$. The constant function $K(n) = 1$ when $n=3$ and $K(n) = \frac{3}{2}$ when $n\geq 4$. Similarly, $\int_{M} \Big\{F^{\prime\prime}(\Phi) \Big[\frac{1}{2}|\nabla\Phi|^{2} - \sum_{\alpha,i,j} \phi_{ij}^{\alpha} \Phi_{j}H_{i}^{\alpha}\Big] + F(\Phi)H^{2} + F^{\prime}(\Phi) (2c-K(n)\Phi)\Phi\Big\} dv \geq 0$, if $F^{\prime}(\Phi) \leq 0$. We also prove the following: If $M^{2}$ is a compact Willmore surface in the $n$-dimensional space form $\mathbb{N}^{n}(c)$. Then $\int_{M}\Phi (C(n)(c+\frac{H^{2}}{2})-\Phi)\leq 0$, where $C(n)=2$ when $n=3$ and $C(n)=\frac{4}{3}$ when $n\geq 4$. If $0\leq\Phi\leq C(n)(c+\frac{H^{2}}{2})$, then either $\Phi=0$ and $M$ is totally umbilical sphere, or $\Phi=C(n)(c+\frac{H^{2}}{2})$. In the latter case, either $M$ is the Clifford torus in $S^{3}$ of $\mathbb{N}^{n}(c)$, or $M$ is the Veronese surface in $S^{4}$ of $\mathbb{N}^{n}(c)$.