Contact stationary Legendrian surfaces in $S^5$

Abstract

Let $(M^5,\alpha,g_\alpha,J)$ be a 5-dimensional Sasakian Einstein manifold with contact 1-form $\alpha$, associated metric $g_\alpha$ and almost complex structure $J$ and $L$ a contact stationary Legendrian surface in $M^5$. We will prove that $L$ satisfies the following equation \begin{eqnarray}\label{equ} -\Delta^\nu H+(K-1)H=0, \end{eqnarray} where $\Delta^\nu$ is the normal Laplacian w.r.t the metric $g$ on $L$ induced from $g_\alpha$ and $K$ is the Gauss curvature of $(L,g)$. Using equation \eqref{equ} and a new Simons' type inequality for Legendrian surfaces in the standard unit sphere $\mathbb{S}^5$, we prove an integral inequality for contact stationary Legendrian surfaces in $\mathbb{S}^5$. In particular, we prove that if $L$ is a contact stationary Legendrian surface in $\mathbb{S}^5$, $B$ is the second fundamental form of $L$, $S=|B|^2$, $\rho^2=S-2H^2$ and $$0\leq S\leq 2,$$ then we have either $\rho^2=0$ and $L$ is totally umbilic or $\rho^2\neq 0$, $S=2, H=0$ and $L$ is a flat minimal Legendrian torus.