Curve shortening flow with an ambient force field
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866915021540294656 |
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| author | Cuthbertson, Sam Wheeler, Glen Wheeler, Valentina-Mira |
| author_facet | Cuthbertson, Sam Wheeler, Glen Wheeler, Valentina-Mira |
| contents | In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the $L^{\infty}$ sense. We prove that closed embedded curves that have a minimum of curvature sufficiently large shrink to round points. The method of proof follows along the same lines of Gage and Hamilton, in that we study a rescaling to prove curvature bounds. We additionally show that the influence of an ambient force field may make such a result untrue, by giving sufficient conditions on the ambient field that ensures eventual non-convexity of an initially convex curve evolving under the flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curve shortening flow with an ambient force field Cuthbertson, Sam Wheeler, Glen Wheeler, Valentina-Mira Analysis of PDEs Differential Geometry 53E40 In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the $L^{\infty}$ sense. We prove that closed embedded curves that have a minimum of curvature sufficiently large shrink to round points. The method of proof follows along the same lines of Gage and Hamilton, in that we study a rescaling to prove curvature bounds. We additionally show that the influence of an ambient force field may make such a result untrue, by giving sufficient conditions on the ambient field that ensures eventual non-convexity of an initially convex curve evolving under the flow. |
| title | Curve shortening flow with an ambient force field |
| topic | Analysis of PDEs Differential Geometry 53E40 |
| url | https://arxiv.org/abs/2411.03579 |