#### Vol. 293, No. 1, 2018

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

### Yong Luo

Vol. 293 (2018), No. 1, 101–120
##### 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 $S=|B{|}^{2}$, ${\rho }^{2}=S-2{H}^{2}$ 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
##### Authors
 Yong Luo School of Mathematics and Statistics Wuhan University Wuhan China