Paper · May 4, 2024

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

Carlo Sinestrari, Liangjun Weng

math.DGWider flowsarXiv:2405.02722Calc. Var. Partial Differential Equations 63 (2024), no. 9, Paper No. 237, 27 pp
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Abstract

In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate that for any convex initial hypersurface with a capillary boundary, the flow exists for all time and smoothly converges to a spherical cap as

Comments: 24 pages, 1 figure

Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsFree boundary & capillaryLong-time existence & convergence

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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)3Weng 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 constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)2Mei 2022 A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane (Xiaoxiang Chai and Yimin Chen, 2024)8Chai 2024 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)5Mei 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)6Mei 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 Free boundary & capillary, Volume-preserving & constrained flows, Other curvature flows · Same author Liangjun Weng · Shared terms “half space”, “spherical cap”

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

    Xinqun Mei, Guofang Wang 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”

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

    Liangjun Weng and Chao XiaWider flows PDF

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

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

    Xinqun Mei and Liangjun WengWider flows PDF

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

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

    Xinqun Mei and Liangjun WengWider flows PDF

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

  6. Uniqueness of capillary Gauss solitons

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

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

  7. Quermassintegral inequalities for capillary hypersurfaces in a horoball

    Jinyu Guo and Liangjun WengWider flows PDF

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

  8. A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane

    Xiaoxiang Chai and Yimin ChenWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “initial hypersurface”

Read around it

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) 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)

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.