Rigidity and Classification of Legendrian Self-Shrinkers

TL;DR

This paper classifies Legendrian self-shrinkers in certain Euclidean spaces and establishes a rigidity theorem showing that under specific conditions, these self-shrinkers are uniquely characterized as flat minimal Legendrian Clifford tori, linking to special Lagrangian cones.

Contribution

It provides the first classification of Legendrian self-shrinkers in $ ^3$ and $ ^5$, and proves a rigidity theorem analogous to Li-Wang's result, identifying unique geometric structures.

Findings

Legendrian self-shrinkers in $ ^3$ and $ ^5$ are classified.

Under certain curvature bounds, the Legendrian self-shrinkers are shown to be flat minimal Legendrian Clifford tori.

The associated cones are identified as Harvey-Lawson special Lagrangian cones.

Abstract

In this article, we first classify Legendrian self-shrinkers in $\mathbb{R}% ^{3}$ and $\mathbb{R}^{5}$. We then proved a Legendrian rigidity theorem, which can be regarded as an analogue of the result of Li-Wang \cite{lw}. More precisely, let $F(\Sigma)\subset\mathbb{R}^{5}$ be an orientable Legendrian self-shrinker, if $\Vert A\Vert_{g}^{2}\leq2$ and the associated Legendrian immersion $\bar{F}\subset\mathbb{R}^{4}\times\mathbb{S}^{1}$ is compact, then $\bar{F}$ must be a flat minimal generalized Legendrian Clifford torus in $\mathbb{S}^{5}$, whose cone $\mathcal{C}(\bar{F}(\Sigma))$ is the Harvey-Lawson special Lagrangian cone in $\mathbb{C}^{3}$.