Monotonicity Formulas for Capillary Surfaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915787190566912 |
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| author | Wang, Guofang Xia, Chao Zhang, Xuwen |
| author_facet | Wang, Guofang Xia, Chao Zhang, Xuwen |
| contents | In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $\mathbb{R}^3_+$ and in the unit ball $\mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221. \href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7}) for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $\mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030. \href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007}) to the capillary setting, which is different to another optimal area estimate proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201. \href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monotonicity Formulas for Capillary Surfaces Wang, Guofang Xia, Chao Zhang, Xuwen Differential Geometry 53C42, 53A10, 49Q15 In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $\mathbb{R}^3_+$ and in the unit ball $\mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221. \href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7}) for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $\mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030. \href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007}) to the capillary setting, which is different to another optimal area estimate proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201. \href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}). |
| title | Monotonicity Formulas for Capillary Surfaces |
| topic | Differential Geometry 53C42, 53A10, 49Q15 |
| url | https://arxiv.org/abs/2409.03314 |