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} -Δ^νH+(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 $$0\le S\le 2,$$ then we have either ${\rho }^2=0$ and $L$ is totally umbilic or ${\rho }^2\ne 0$, $S=2,H=0$ and $L$ is a flat minimal Legendrian torus.

Errors corrected. An appendix added

References in corpus (1)

• Hamiltonian stationary tori in the complex projective plane

Cited by in corpus (1)

• On energy gap phenomena of the Whitney sphere and related problems