Contact stationary Legendrian surfaces in $\mathbb{S}^5$

Yong Luo

Vol. 293 (2018), No. 1, 101–120

DOI: 10.2140/pjm.2018.293.101

Abstract

Let $\left(M^5,\alpha ,g_{\alpha },J\right)$ be a 5-dimensional Sasakian Einstein manifold with contact 1-form $\alpha$, associated metric $g_{\alpha }$ and almost complex structure $J$, and let $L$ be a contact stationary Legendrian surface in $M^5$. We will prove that $L$ satisfies the equation

$$-{\Delta }^{\nu }H+\left(K-1\right)H=0,$$

where ${\Delta }^{\nu }$ is the normal Laplacian with respect to the metric $g$ on $L$ induced from $g_{\alpha }$ and $K$ is the Gauss curvature of $\left(L,g\right)$.

Using this equation 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 ${\mathbb{S}}^5$ and $B$ is the second fundamental form of $L$, with $S=|B{|}^2$, ${\rho }^2=S-2H^2$ 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.

Keywords

volume functional, Simons inequality, gap theorem

Mathematical Subject Classification 2010

Primary: 53B25, 53C42

Milestones

Received: 17 November 2016

Accepted: 11 September 2017

Published: 3 November 2017