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

## TL;DR

This paper classifies Willmore Legendrian surfaces in the 5-sphere, establishes integral inequalities, and introduces a flow for contact stationary Legendrian Willmore surfaces in Sasakian manifolds, especially $S^5$.

## Contribution

It provides a classification of Willmore Legendrian spheres, proves integral bounds, and develops a new flow for contact stationary Legendrian Willmore surfaces in Sasakian manifolds.

## Key findings

- Classification of Willmore Legendrian spheres in $S^5$.
- Integral inequality for Willmore Legendrian surfaces with bounds on second fundamental form.
- Existence and well-posedness of a higher order flow in Sasakian Einstein manifolds.

## Abstract

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 $\mathbb{S}^5$.