Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.15279 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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 $F(Σ)\subset\mathbb{R}^{5}$ 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 $\mathcal{C}(\bar{F}(Σ))$ is the Harvey-Lawson special Lagrangian cone in $\mathbb{C}^{3}$.