Paper · Oct 16, 2019

Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball

Guofang Wang, Chao Xia

math.DGWider flowsarXiv:1910.07253v2
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Abstract

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

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Submitted
Oct 16, 2019
Last revised
Jun 5, 2020 (v2)
Subjects
math.DG

Comments: 13 pages, final version, Comm. Anal. Geom., to appear

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

Its neighbourhood

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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 2016, 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 A geometric inequality for convex free boundary hypersurfaces in the unit ball (Ben Lambert and Julian Scheuer, 2016)5Lambert 2016 Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries (Kunbo Wang, 2017)6Wang 2017 Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball (Julian Scheuer, Guofang Wang and Chao Xia, 2018)1Scheuer 2018 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)2Wang 2022 A constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)3Mei 2022 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)8Sinestrari 2024 A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane (Xiaoxiang Chai and Yimin Chen, 2024)7Chai 2024
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 with free boundary in a ball

    Julian Scheuer, Guofang Wang and Chao XiaWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same authors Guofang Wang, Chao Xia · Shared terms “boundary hypersurfaces”

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

    Guofang Wang, Liangjun Weng and Chao XiaWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same authors Guofang Wang, Chao Xia · Shared terms “spherical cap”

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

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Same author Guofang Wang · Shared terms “star shaped”, “spherical cap”

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

    Liangjun Weng and Chao XiaWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same author Chao Xia

  5. A geometric inequality for convex free boundary hypersurfaces in the unit ball

    Ben Lambert and Julian ScheuerWider flows

    Topics Free boundary & capillary · Shared terms “boundary hypersurfaces”

  6. Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

    Kunbo WangWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence

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

    Xiaoxiang Chai and Yimin ChenWider flows

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

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

    Carlo Sinestrari and Liangjun WengWider flows

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

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.

Foundational paper in the Free boundary & capillary guide: Wang–Xia: a Guan–Li type flow for free-boundary hypersurfaces in a ball.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. 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.