Asymptotic circularity of immortal area-preserving curvature flows
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909393455415296 |
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| author | Miura, Tatsuya |
| author_facet | Miura, Tatsuya |
| contents | For a class of area-preserving curvature flows of closed planar curves, we prove that every immortal solution becomes asymptotically circular without any additional assumptions on initial data. As a particular corollary, every solution of zero enclosed area blows up in finite time. This settles an open problem posed by Escher--Ito in 2005 for Gage's area-preserving curve shortening flow, and moreover extends it to the surface diffusion flow of arbitrary order. We also establish a general existence theorem for nontrivial immortal solutions under almost circularity and rotational symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic circularity of immortal area-preserving curvature flows Miura, Tatsuya Differential Geometry Analysis of PDEs 53E40, 53E10, 53A04, 35K55, 35B40, 35B44 For a class of area-preserving curvature flows of closed planar curves, we prove that every immortal solution becomes asymptotically circular without any additional assumptions on initial data. As a particular corollary, every solution of zero enclosed area blows up in finite time. This settles an open problem posed by Escher--Ito in 2005 for Gage's area-preserving curve shortening flow, and moreover extends it to the surface diffusion flow of arbitrary order. We also establish a general existence theorem for nontrivial immortal solutions under almost circularity and rotational symmetry. |
| title | Asymptotic circularity of immortal area-preserving curvature flows |
| topic | Differential Geometry Analysis of PDEs 53E40, 53E10, 53A04, 35K55, 35B40, 35B44 |
| url | https://arxiv.org/abs/2410.06183 |