Paper · Aug 15, 2017

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

Maria Chiara Bertini, Carlo Sinestrari

math.DGWider flowsarXiv:1708.04573v3
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Abstract

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the -th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.

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Submitted
Aug 15, 2017
Last revised
Feb 8, 2018 (v3)
Subjects
math.DG

Comments: 17 pages

Other curvature flowsVolume-preserving & constrained flowsCurvature pinchingScalar curvatureLong-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 Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes (Giuseppe Pipoli and Carlo Sinestrari, 2016)6Pipoli 2016 Volume preserving non homogeneous mean curvature flow of convex hypersurfaces (Maria Chiara Bertini and Carlo Sinestrari, 2016)1Bertini 2016 Mixed volume preserving flow by powers of homogeneous curvature functions of degree one (Shunzi Guo, 2016)2Guo 2016 Volume preserving non homogeneous mean curvature flow in hyperbolic space (Maria Chiara Bertini and Giuseppe Pipoli, 2016)4Bertini 2016 Contracting axially symmetric hypersurfaces by powers of the σₖ-curvature (Haizhong Li, Xianfeng Wang and Jing Wu, 2019)5Li 2019 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)7Wei 2022 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)3Sinestrari 2024 Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces (Carlo Sinestrari and Jacopo Tenan, 2025)8Sinestrari 2025
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 · Same authors Maria Chiara Bertini, Carlo Sinestrari · Shared terms “convex hypersurfaces”, “hypersurfaces converge”, “round sphere”

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

    Shunzi GuoWider flows

    Topics Curvature pinching, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “hypersurfaces converge”, “convex hypersurfaces”, “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, Long-time existence & convergence, Other curvature flows · Same author Carlo Sinestrari

  4. Volume preserving non homogeneous mean curvature flow in hyperbolic space

    Maria Chiara Bertini and Giuseppe PipoliWider flows

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

  5. Contracting axially symmetric hypersurfaces by powers of the -curvature

    Haizhong Li, Xianfeng Wang and Jing WuWider flows

    Topics Curvature pinching, Long-time existence & convergence, Other curvature flows · Shared terms “principal curvatures”, “elementary symmetric”, “hypersurfaces converge”

  6. Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

    Giuseppe Pipoli and Carlo SinestrariWider flows

    Topics Curvature pinching · Same author Carlo Sinestrari · Shared terms “principal curvatures”

  7. 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 “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, Long-time existence & convergence · Same author Carlo Sinestrari · Shared terms “converges smoothly”

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.

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.