Paper · Aug 25, 2024

A fully nonlinear locally constrained curvature flow for capillary hypersurface

Xinqun Mei, Liangjun Weng

math.APWider flowsarXiv:2408.13815Ann. Global Anal. Geom. 67 (2025), no. 1, 5
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Abstract

In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in. As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov-Fenchel inequalities.

Comments: 16 pages

Other curvature flowsVolume-preserving & constrained flowsFree boundary & capillaryLong-time existence & convergenceGeometric 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)5Weng 2021 Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary (Guofang Wang, Liangjun Weng and Chao Xia, 2022)3Wang 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)7Hu 2022 A constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)1Mei 2022 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)2Mei 2024 The capillary Gauss curvature flow (Xinqun Mei, Guofang Wang and Liangjun Weng, 2025)8Mei 2025 Uniqueness of capillary Gauss solitons (Xinqun Mei, Guofang Wang and Liangjun Weng, 2026)6Mei 2026 Quermassintegral inequalities for capillary hypersurfaces in a horoball (Jinyu Guo and Liangjun Weng, 2026)4Guo 2026
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  1. 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 Xinqun Mei, Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “locally constrained”

  2. 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 authors Xinqun Mei, Liangjun Weng · Shared terms “capillary hypersurfaces”, “Alexandrov Fenchel”, “Fenchel inequalities”

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

    Guofang Wang, Liangjun Weng and Chao XiaWider flows PDF

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

  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 “capillary hypersurfaces”, “convex capillary”, “locally constrained”

  5. 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”, “locally constrained”

  6. 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”, “spherical cap”

  7. 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 “capillary hypersurfaces”, “convex capillary”, “half space”

  8. The capillary Gauss curvature flow

    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”, “counterpart”

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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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)