Paper · Mar 16, 2022

Inverse mean curvature flow with a free boundary in hyperbolic space

Xiaoxiang Chai

math.DGWider flowsarXiv:2203.08467
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Abstract

We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.

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Submitted
Mar 16, 2022
Subjects
math.DG, math.AP

Comments: 28 pages, comments welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowSingularitiesFree boundary & capillary

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 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 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)8Bourni 2022 A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane (Xiaoxiang Chai and Yimin Chen, 2024)1Chai 2024 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)6Mei 2024 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)4Gao 2024 Evolution of the Torsional Rigidity under Geometric Flows (Vicent Gimeno i Garcia and Fernán González-Ibáñez, 2024)7Garcia 2024 Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds (Liam Mazurowski and Jintian Zhu, 2025)3Mazurowski 2025 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)2Pan 2025 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)5Ghosh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane

    Xiaoxiang Chai and Yimin ChenWider flows

    Topics Free boundary & capillary · Same author Xiaoxiang Chai · Shared terms “totally geodesic”, “supported”, “hyperbolic space”

  2. Inverse curvature flows for capillary hypersurfaces in the unit ball

    Shujing Pan and Bo YangWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “convex hypersurfaces”

  3. Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

    Liam Mazurowski and Jintian ZhuWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows

  4. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “hypersurface inside”, “convex hypersurfaces”

  5. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows

    Topics Free boundary & capillary, Singularities

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

    Xinqun Mei and Liangjun WengWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “hyperbolic space”

  7. Evolution of the Torsional Rigidity under Geometric Flows

    Vicent Gimeno i Garcia and Fernán González-Ibáñez

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows

  8. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows

    Topics Free boundary & capillary

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) 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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