Paper · Jun 11, 2024

Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

Xinqun Mei, Liangjun Weng

math.DGWider flowsarXiv:2406.07304v2
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Abstract

In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By constructing a new locally constrained inverse curvature flow, we obtain the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space. This generalizes a theorem of Brendle-Guan-Li for convex closed hypersurfaces in hyperbolic space.

Comments: Minor corrections. 35 pages, 1 figure

Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryGeometric inequalities

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball (Liangjun Weng and Chao Xia, 2021)4Weng 2021 Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary (Guofang Wang, Liangjun Weng and Chao Xia, 2022)1Wang 2022 A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space (Yingxiang Hu, Yong Wei, Bo Yang et al., 2022)6Hu 2022 A constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)2Mei 2022 A fully nonlinear locally constrained curvature flow for capillary hypersurface (Xinqun Mei and Liangjun Weng, 2024)3Mei 2024 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)7Pan 2025 Uniqueness of capillary Gauss solitons (Xinqun Mei, Guofang Wang and Liangjun Weng, 2026)8Mei 2026 Quermassintegral inequalities for capillary hypersurfaces in a horoball (Jinyu Guo and Liangjun Weng, 2026)5Guo 2026
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  1. Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

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    Topics Geometric inequalities, Free boundary & capillary, Volume-preserving & constrained flows · Same author Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “capillary hypersurfaces”

  2. A constrained mean curvature flow and Alexandrov-Fenchel inequalities

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    Topics Geometric inequalities, Free boundary & capillary, Volume-preserving & constrained flows · Same authors Xinqun Mei, Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “locally constrained”

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    Topics Geometric inequalities, Free boundary & capillary, Volume-preserving & constrained flows · Same authors Xinqun Mei, Liangjun Weng · Shared terms “capillary hypersurfaces”, “Alexandrov Fenchel”, “Fenchel inequalities”

  4. Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

    Liangjun Weng and Chao XiaWider flows PDF

    Topics Geometric inequalities, Free boundary & capillary, Volume-preserving & constrained flows · Same author Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “quermassintegral”

  5. Quermassintegral inequalities for capillary hypersurfaces in a horoball

    Jinyu Guo and Liangjun WengWider flows PDF

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  6. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

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    Topics Geometric inequalities, Free boundary & capillary, Inverse mean curvature flow · Shared terms “capillary hypersurfaces”, “Alexandrov Fenchel”, “Fenchel inequalities”

  7. Inverse curvature flows for capillary hypersurfaces in the unit ball

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    Topics Geometric inequalities, Free boundary & capillary, Inverse mean curvature flow · Shared terms “capillary hypersurfaces”, “Alexandrov Fenchel”, “Fenchel inequalities”

  8. Uniqueness of capillary Gauss solitons

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

    Topics Free boundary & capillary, Other curvature flows · Same authors Xinqun Mei, Liangjun Weng · Shared terms “capillary hypersurfaces”, “convex capillary”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)