Paper · Jun 9, 2022

Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

Guofang Wang, Liangjun Weng, Chao Xia

math.DGWider flowsarXiv:2206.04639v2Math. Ann. 388 (2024), no. 2, 2121--2154
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Abstract

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov-Fenchel inequalities. In order to prove these results, we construct a new locally constrained curvature flow and prove that the flow converges globally to a spherical cap.

Comments: Final version, to appear in Mathematische Annalen

Other curvature flowsVolume-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)1Weng 2021 A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space (Yingxiang Hu, Yong Wei, Bo Yang et al., 2022)5Hu 2022 A constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)2Mei 2022 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)8Sinestrari 2024 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)3Mei 2024 A fully nonlinear locally constrained curvature flow for capillary hypersurface (Xinqun Mei and Liangjun Weng, 2024)4Mei 2024 Uniqueness of capillary Gauss solitons (Xinqun Mei, Guofang Wang and Liangjun Weng, 2026)7Mei 2026 Quermassintegral inequalities for capillary hypersurfaces in a horoball (Jinyu Guo and Liangjun Weng, 2026)6Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 authors Liangjun Weng, Chao Xia · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “convex hypersurfaces”

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

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

    Topics Geometric inequalities, Free boundary & capillary, Volume-preserving & constrained flows · Same authors Guofang Wang, Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “half space”

  3. Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

    Xinqun Mei and Liangjun WengWider flows PDF

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

  4. A fully nonlinear locally constrained curvature flow for capillary hypersurface

    Xinqun Mei and Liangjun WengWider flows PDF

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

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

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows PDF

    Topics Geometric inequalities, Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “half space”, “Alexandrov Fenchel”, “Fenchel inequalities”

  6. Quermassintegral inequalities for capillary hypersurfaces in a horoball

    Jinyu Guo and Liangjun WengWider flows PDF

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

  7. Uniqueness of capillary Gauss solitons

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

    Topics Free boundary & capillary, Other curvature flows · Same authors Guofang Wang, Liangjun Weng · Shared terms “convex capillary”, “half space”, “spherical cap”

  8. Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

    Carlo Sinestrari and Liangjun WengWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Other curvature flows · Same author Liangjun Weng · Shared terms “half space”, “spherical cap”

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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) 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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)