Paper · Oct 24, 2016

Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

Maria Chiara Bertini, Carlo Sinestrari

math.DGWider flowsarXiv:1610.07436
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As on arXiv

Abstract

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.

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Submitted
Oct 24, 2016
Subjects
math.DG
Mean curvature flowVolume-preserving & constrained flowsHomogeneous spacesGeometric inequalities

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 Mixed volume preserving flow by powers of homogeneous curvature functions of degree one (Shunzi Guo, 2016)6Guo 2016 Volume preserving non homogeneous mean curvature flow in hyperbolic space (Maria Chiara Bertini and Giuseppe Pipoli, 2016)1Bertini 2016 Volume preserving flow by powers of symmetric polynomials in the principal curvatures (Maria Chiara Bertini and Carlo Sinestrari, 2017)2Bertini 2017 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)7Andrews 2018 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)3Sinestrari 2024 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)4Wei 2024 Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces (Carlo Sinestrari and Jacopo Tenan, 2025)8Sinestrari 2025 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)5Cabezas-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 in hyperbolic space

    Maria Chiara Bertini and Giuseppe PipoliWider flows

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

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

    Maria Chiara Bertini and Carlo SinestrariWider flows

    Topics Volume-preserving & constrained flows · Same authors Maria Chiara Bertini, Carlo Sinestrari · Shared terms “convex hypersurfaces”, “hypersurfaces converge”, “round sphere”

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

    Carlo Sinestrari and Liangjun WengWider flows

    Topics Volume-preserving & constrained flows · Same author Carlo Sinestrari · Shared terms “area preserving”

  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 · Shared terms “nonhomogeneous”, “speed function”, “convex hypersurfaces”

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

    Esther Cabezas-Rivas and Carlo SinestrariWider flows

    Topics Volume-preserving & constrained flows · Same author Carlo Sinestrari

  6. Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

    Shunzi GuoWider flows

    Topics Volume-preserving & constrained flows, Homogeneous spaces · Shared terms “convex hypersurfaces”, “hypersurfaces converge”, “round sphere”

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

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Homogeneous spaces · Shared terms “convex hypersurfaces”

  8. Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces

    Carlo Sinestrari and Jacopo TenanWider flows

    Topics Volume-preserving & constrained flows · Same author Carlo Sinestrari

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) 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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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.