A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866915464478720000 |
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| author | Hu, Yingxiang Wei, Yong Yang, Bo Zhou, Tailong |
| author_facet | Hu, Yingxiang Wei, Yong Yang, Bo Zhou, Tailong |
| contents | In this paper, we study the locally constrained inverse curvature flow for hypersurfaces in the half-space with $θ$-capillary boundary, which was recently introduced by Wang-Weng-Xia. Assume that the initial hypersurface is strictly convex with the contact angle $θ\in (0,π/2]$. We prove that the solution of the flow remains to be strictly convex for $t>0$, exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space with the contact angle $θ\in(0,π/2]$. Along the proof, we develop a new tensor maximum principle for parabolic equations on compact manifold with proper Neumann boundary condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_12479 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space Hu, Yingxiang Wei, Yong Yang, Bo Zhou, Tailong Differential Geometry Analysis of PDEs 53C44, 53C21, 35K93, 52A40 In this paper, we study the locally constrained inverse curvature flow for hypersurfaces in the half-space with $θ$-capillary boundary, which was recently introduced by Wang-Weng-Xia. Assume that the initial hypersurface is strictly convex with the contact angle $θ\in (0,π/2]$. We prove that the solution of the flow remains to be strictly convex for $t>0$, exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space with the contact angle $θ\in(0,π/2]$. Along the proof, we develop a new tensor maximum principle for parabolic equations on compact manifold with proper Neumann boundary condition. |
| title | A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space |
| topic | Differential Geometry Analysis of PDEs 53C44, 53C21, 35K93, 52A40 |
| url | https://arxiv.org/abs/2209.12479 |