A delayed interior area-to-height estimate for the Curve Shortening Flow
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910090562371584 |
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| author | Sobnack, Arjun |
| author_facet | Sobnack, Arjun |
| contents | The principle of delayed parabolic regularity for the Curve Shortening Flow - that if two evolving curves bound a region of area $\mathcal A$, then, starting from time ${\mathcal A}/π$, the regularity of one curve is controllable in terms of the time elapsed, the area $\mathcal A$ and the regularity of the other curve - was proposed by Topping & the author in (Sobnack & Topping, 2024), where they also provided a number of graphical situations in which their delayed regularity framework is valid. In this paper, we generalise some of the results in (Sobnack & Topping, 2024) within the graphical setting, ultimately by showing that there holds an interior graphical estimate for the Curve Shortening Flow in the spirit of the proposed framework. We also provide a few applications of our estimate, such as the existence of Graphical Curve Shortening Flows starting weakly from Radon measures without point masses. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_16581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A delayed interior area-to-height estimate for the Curve Shortening Flow Sobnack, Arjun Analysis of PDEs Differential Geometry 35B45, 35B65, 35K55, 35K20, 53E10 The principle of delayed parabolic regularity for the Curve Shortening Flow - that if two evolving curves bound a region of area $\mathcal A$, then, starting from time ${\mathcal A}/π$, the regularity of one curve is controllable in terms of the time elapsed, the area $\mathcal A$ and the regularity of the other curve - was proposed by Topping & the author in (Sobnack & Topping, 2024), where they also provided a number of graphical situations in which their delayed regularity framework is valid. In this paper, we generalise some of the results in (Sobnack & Topping, 2024) within the graphical setting, ultimately by showing that there holds an interior graphical estimate for the Curve Shortening Flow in the spirit of the proposed framework. We also provide a few applications of our estimate, such as the existence of Graphical Curve Shortening Flows starting weakly from Radon measures without point masses. |
| title | A delayed interior area-to-height estimate for the Curve Shortening Flow |
| topic | Analysis of PDEs Differential Geometry 35B45, 35B65, 35K55, 35K20, 53E10 |
| url | https://arxiv.org/abs/2502.16581 |