Rigidity and Classification of Legendrian Self-Shrinkers

[2508.15279] Rigidity and Classification of Legendrian Self-Shrinkers Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2508.15279 (math) [Submitted on 21 Aug 2025 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Rigidity and Classification of Legendrian Self-Shrinkers Authors: Shu-Cheng Chang , Chin-Tung Wu , Liuyang Zhang , Qiuxia Zhang View PDF HTML (experimental) 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 $\mathbb{C}^{3}$. Subjects: Differential Geometry (math.DG) MSC classes: 53C44, 53C56 Cite as: arXiv:2508.15279 [math.DG] (or arXiv:2508.15279v2 [math.DG] for this version) https://doi.org/10.48550/arXiv.2508.15279 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Liuyang Zhang [ view email ] [v1] Thu, 21 Aug 2025 06:11:53 UTC (20 KB) [v2] Fri, 28 Aug 2026 01:07:21 UTC (22 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2025-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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