Rigidity and Classification of Legendrian Self-Shrinkers

In this article, we first classify Legendrian self-shrinkers in $\mathbb{R}$ 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 $F(\mathrm{\Sigma })\subset {\mathbb{R}}^5$ be an orientable Legendrian self-shrinker, if $\mathrm{\parallel }A{\mathrm{\parallel }}_g^2\le 2$ 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 $\mathcal{C}(\bar{F}(\mathrm{\Sigma }))$ is the Harvey-Lawson special Lagrangian cone in ${\mathbb{C}}^3$.