Willmore-type inequality in unbounded convex sets
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910858910629888 |
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| author | Jia, Xiaohan Wang, Guofang Xia, Chao Zhang, Xuwen |
| author_facet | Jia, Xiaohan Wang, Guofang Xia, Chao Zhang, Xuwen |
| contents | In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $K\subset\mathbb{R}^{n+1}$ $(n\ge 2)$, for any embedded hypersurface $Σ\subset K$ with boundary $\partialΣ\subset \partial K$ satisfying a certain contact angle condition, there holds $$\frac1{n+1}\int_Σ\vert{H}\vert^n{\rm d}A\ge{\rm AVR}(K)\vert\mathbb{B}^{n+1}\vert.$$ Moreover, equality holds if and only if $Σ$ is a part of a sphere and $K\setminusΩ$ is a part of the solid cone determined by $Σ$. Here $Ω$ is the bounded domain enclosed by $Σ$ and $\partial K$, $H$ is the normalized mean curvature of $Σ$, and ${\rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Willmore-type inequality in unbounded convex sets Jia, Xiaohan Wang, Guofang Xia, Chao Zhang, Xuwen Differential Geometry 53C42, 53C20 In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $K\subset\mathbb{R}^{n+1}$ $(n\ge 2)$, for any embedded hypersurface $Σ\subset K$ with boundary $\partialΣ\subset \partial K$ satisfying a certain contact angle condition, there holds $$\frac1{n+1}\int_Σ\vert{H}\vert^n{\rm d}A\ge{\rm AVR}(K)\vert\mathbb{B}^{n+1}\vert.$$ Moreover, equality holds if and only if $Σ$ is a part of a sphere and $K\setminusΩ$ is a part of the solid cone determined by $Σ$. Here $Ω$ is the bounded domain enclosed by $Σ$ and $\partial K$, $H$ is the normalized mean curvature of $Σ$, and ${\rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space. |
| title | Willmore-type inequality in unbounded convex sets |
| topic | Differential Geometry 53C42, 53C20 |
| url | https://arxiv.org/abs/2409.03321 |