Paper · Nov 10, 2021

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

Liangjun Weng, Chao Xia

math.DGWider flowsarXiv:2111.05773v3Trans. Amer. Math. Soc. 375 (2022), no. 12, 8851--8883
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Abstract

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the -th quermassintegral and non-decreases the -th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in . This generalizes the result in for convex hypersurfaces with free boundary in .

Comments: To appear in Trans. Amer. Math. Soc

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 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)8Hu 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)7Sinestrari 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)5Mei 2024 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)6Pan 2025 Quermassintegral inequalities for capillary hypersurfaces in a horoball (Jinyu Guo and Liangjun Weng, 2026)4Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Guofang Wang, 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”

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    Xinqun Mei, Guofang Wang 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”, “convex hypersurfaces”

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

  4. 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 “quermassintegral”, “constrained nonlinear”, “nonlinear curvature”

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

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    Topics Geometric inequalities, Free boundary & capillary, Other curvature flows · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “unit Euclidean”

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    Topics Free boundary & capillary, Volume-preserving & constrained flows, Other curvature flows · Same author Liangjun Weng

  8. 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 “Alexandrov Fenchel”, “Fenchel inequalities”, “locally constrained”

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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)