Paper · Jul 16, 2025

Inverse curvature flows for capillary hypersurfaces in the unit ball

Shujing Pan, Bo Yang

math.DGWider flowsarXiv:2507.12097
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Abstract

In this paper, we study inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball. We establish the existence and convergence results for a class of such flows. As an application, we derive a family of Alexandrov Fenchel inequalities for weakly convex hypersurfaces with free boundary.

Comments: 28 pages. All comments are welcome

Other curvature flowsInverse mean curvature flowFree boundary & capillaryGeometric inequalities

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Connected papers

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)6Wang 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)1Hu 2022 The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces (Shujing Pan and Bo Yang, 2023)7Pan 2023 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)5Pan 2024 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)3Mei 2024 Capillary quermassintegral inequalities in the unit ball (Shujing Pan and Julian Scheuer, 2026)2Pan 2026 Quermassintegral inequalities for capillary hypersurfaces in a horoball (Jinyu Guo and Liangjun Weng, 2026)8Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Inverse mean curvature flow · Same author Bo Yang · Shared terms “capillary hypersurfaces”, “Alexandrov Fenchel”, “Fenchel inequalities”

  2. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows PDF

    Topics Geometric inequalities, Free boundary & capillary, Other curvature flows · Same author Shujing Pan · Shared terms “unit ball”, “capillary hypersurfaces”, “Euclidean”

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

    Xinqun Mei and Liangjun WengWider flows PDF

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

  5. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows PDF

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same authors Shujing Pan, Bo Yang

  6. 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, Other curvature flows · Shared terms “capillary hypersurfaces”, “Alexandrov Fenchel”, “Fenchel inequalities”

  7. The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

    Shujing Pan and Bo YangWider flows PDF

    Topics Geometric inequalities, Other curvature flows · Same authors Shujing Pan, Bo Yang

  8. Quermassintegral inequalities for capillary hypersurfaces in a horoball

    Jinyu Guo and Liangjun WengWider flows PDF

    Topics Geometric inequalities, Free boundary & capillary, Other curvature flows · Shared terms “capillary hypersurfaces”

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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) 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)