The capillary Gauss curvature flow
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913889368670208 |
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| author | Mei, Xinqun Wang, Guofang Weng, Liangjun |
| author_facet | Mei, Xinqun Wang, Guofang Weng, Liangjun |
| contents | In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The capillary Gauss curvature flow Mei, Xinqun Wang, Guofang Weng, Liangjun Differential Geometry Analysis of PDEs Primary: 53C21, 35K55. Secondary: 52A20, 35B65, 35C08 In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture. |
| title | The capillary Gauss curvature flow |
| topic | Differential Geometry Analysis of PDEs Primary: 53C21, 35K55. Secondary: 52A20, 35B65, 35C08 |
| url | https://arxiv.org/abs/2506.09840 |