On Willmore Legendrian surfaces in $\mathbb{S}^5$ and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in \cite{Luo} to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in $\mathbb{S}^5$, and then we use this relation to prove a classification result for Willmore Legendrian spheres in $\mathbb{S}^5$. We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in $\mathbb{S}^5$ belongs to $[0,2]$, then it must either be $0$ and $L$ is totally geodesic or $2$ and $L$ is a flat minimal Legendrian tori, which generalizes a result of \cite{YKM}. We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let $\Sigma$ be a closed surface and $(M,\alpha,g_\alpha,J)$ a 5-dimensional Sasakian manifold with a contact form $\alpha$, an associated metric $g_\alpha$ and an almost complex structure $J$. Assume that $f:\Sigma\mapsto M$ is a Legendrian immersion. Then $f$ is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if $(M,\alpha,g_\alpha,J)$ is a Sasakian Einstein manifold, in particular $\mathbb{S}^5$.