Let
$\left({M}^{5},\alpha ,{g}_{\alpha},J\right)$
be a 5dimensional Sasakian Einstein manifold with contact 1form
$\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
$${\Delta}^{\nu}H+\left(K1\right)H=0,$$
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
$S=B{}^{2}$,
${\rho}^{2}=S2{H}^{2}$
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.
