Star-shaped Curves under Gage's Area-preserving Flow and the CSF
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915078106775552 |
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| author | Gao, Laiyuan Zhang, Shicheng Zhang, Yuntao |
| author_facet | Gao, Laiyuan Zhang, Shicheng Zhang, Yuntao |
| contents | Mayer asks a question what closed, embedded and nonconvex initial curves guarantee that Gage's area-preserving flow (GAPF) exists globally. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this paper, we prove this conjecture by using Dittberner's singularity analysis theory. A star-shaped ``flying wing" curve is constructed to show that GAPF may not always preserve the star-shapedness of evolving curves. This example is also a negative answer to Mantegazza's open problem whether the curve shortening flow (CSF) always preserves the star shape of the evolving curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18102 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Star-shaped Curves under Gage's Area-preserving Flow and the CSF Gao, Laiyuan Zhang, Shicheng Zhang, Yuntao Differential Geometry 35C44, 35K05, 53A04 Mayer asks a question what closed, embedded and nonconvex initial curves guarantee that Gage's area-preserving flow (GAPF) exists globally. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this paper, we prove this conjecture by using Dittberner's singularity analysis theory. A star-shaped ``flying wing" curve is constructed to show that GAPF may not always preserve the star-shapedness of evolving curves. This example is also a negative answer to Mantegazza's open problem whether the curve shortening flow (CSF) always preserves the star shape of the evolving curves. |
| title | Star-shaped Curves under Gage's Area-preserving Flow and the CSF |
| topic | Differential Geometry 35C44, 35K05, 53A04 |
| url | https://arxiv.org/abs/2412.18102 |