Paper · Sep 27, 2022

A constrained mean curvature flow and Alexandrov-Fenchel inequalities

Xinqun Mei, Guofang Wang, Liangjun Weng

math.DGWider flowsarXiv:2209.13463v3Int. Math. Res. Not. IMRN 2024, no. 1, 152--174
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Abstract

In this article, we study a locally constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary in the half-space. We prove its long-time existence and the global convergence to a spherical cap. Furthermore, the capillary quermassintegrals defined in evolve monotonically along the flow, and hence we establish a class of new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space.

Comments: Final version, to appear in Int. Math. Res. Not. IMRN

Mean curvature flowVolume-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)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)5Hu 2022 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)6Sinestrari 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)2Mei 2024 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)7Guo 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 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 authors Guofang Wang, Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “half space”

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

  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 authors Xinqun Mei, Liangjun Weng · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “locally constrained”

  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”, “convex hypersurfaces”

  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 “Alexandrov Fenchel”, “Fenchel inequalities”, “half space”

  6. 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, Long-time existence & convergence · Same author Liangjun Weng · Shared terms “half space”, “spherical cap”

  7. 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”, “locally constrained”, “monotonically”

  8. Uniqueness of capillary Gauss solitons

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

    Topics Free boundary & capillary · Same authors Xinqun Mei, Guofang Wang, Liangjun Weng · Shared terms “spherical cap”, “half space”

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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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.