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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2603.09112 |
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| _version_ | 1866917327997501440 |
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| author | Choi, Kyeongsu Seo, Dong-Hwi Su, Wei-Bo Zhao, Kai-Wei |
| author_facet | Choi, Kyeongsu Seo, Dong-Hwi Su, Wei-Bo Zhao, Kai-Wei |
| contents | We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone.
An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together $m$ translating grim reaper curves. For each $m$, there exists a $(2m-1)$-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You.
In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of ancient finite-entropy curve shortening flows Choi, Kyeongsu Seo, Dong-Hwi Su, Wei-Bo Zhao, Kai-Wei Differential Geometry We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together $m$ translating grim reaper curves. For each $m$, there exists a $(2m-1)$-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval. |
| title | Classification of ancient finite-entropy curve shortening flows |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2603.09112 |