Paper · Dec 15, 2016

Volume preserving non homogeneous mean curvature flow in hyperbolic space

Maria Chiara Bertini, Giuseppe Pipoli

math.DGWider flowsarXiv:1612.05288v2
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Abstract

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is ispired by [10] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour of a suitable ratio associated to the hypersurface that converges exponentially in time to the value associated to a geodesic sphere.

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Submitted
Dec 15, 2016
Last revised
Jan 23, 2017 (v2)
Subjects
math.DG

Comments: 15 pages

Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsHomogeneous spacesLong-time existence & convergence

Its neighbourhood

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 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 Volume preserving non homogeneous mean curvature flow of convex hypersurfaces (Maria Chiara Bertini and Carlo Sinestrari, 2016)1Bertini 2016 Volume preserving flow by powers of symmetric polynomials in the principal curvatures (Maria Chiara Bertini and Carlo Sinestrari, 2017)3Bertini 2017 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)7Andrews 2018 Nonhomogeneous expanding flows in hyperbolic spaces (Giuseppe Pipoli, 2020)6Pipoli 2020 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)2Huang 2021 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)5Wei 2022 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)4Wei 2024 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)8Cabezas-Riva 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

    Maria Chiara Bertini and Carlo SinestrariWider flows

    Topics Volume-preserving & constrained flows, Homogeneous spaces · Same author Maria Chiara Bertini · Shared terms “principle argument”, “uniform bounds”, “area preserving”

  2. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

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

  3. Volume preserving flow by powers of symmetric polynomials in the principal curvatures

    Maria Chiara Bertini and Carlo SinestrariWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Same author Maria Chiara Bertini

  4. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Volume-preserving & constrained flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “hyperbolic space”, “geodesic sphere”

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

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “hyperbolic space”, “geodesic sphere”

  6. Nonhomogeneous expanding flows in hyperbolic spaces

    Giuseppe PipoliWider flows

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Same author Giuseppe Pipoli · Shared terms “hyperbolic space”

  7. Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Volume-preserving & constrained flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “hyperbolic space”, “geodesic sphere”, “increasing”

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

    Esther Cabezas-Rivas and Carlo SinestrariWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “geodesic sphere”, “exponential convergence”, “hyperbolic space”

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 in the L² sense. 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.