Paper · Jul 28, 2021

Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

Zheng Huang, Longzhi Lin, Zhou Zhang

math.DGWider flowsarXiv:2107.13127
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Abstract

We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the -sense, then the flow exists for all time and converges exponentially to an umbilical sphere.

Comments: 20 pages

Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessLong-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 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)5Laux 2022 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)4Wei 2022 The quermassintegral preserving mean curvature flow in the sphere (Esther Cabezas-Rivas and Julian Scheuer, 2022)6Cabezas-Riva 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)1De Gennaro 2023 A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane (Xiaoxiang Chai and Yimin Chen, 2024)7Chai 2024 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)2Bonforte 2024 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)8Arya 2026 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)3Cabezas-Riva 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Stability & uniqueness

  2. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “Euclidean space”

  3. Geometric -stability of geodesic spheres under volume-preserving mean curvature flow

    Esther Cabezas-Rivas and Carlo SinestrariWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Stability & uniqueness · Shared terms “initial hypersurface”, “hyperbolic space”

  4. Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “hyperbolic space”, “initial hypersurface”

  5. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows PDF

    Topics Volume-preserving & constrained flows, Stability & uniqueness

  6. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “initial hypersurface”, “strictly convex”, “flow exists”

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

    Xiaoxiang Chai and Yimin ChenWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “umbilical”, “enclosed volume”, “hyperbolic space”

  8. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “converges exponentially”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.