Paper · Oct 26, 2016

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

Shunzi Guo

math.DGWider flowsarXiv:1610.08214
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Abstract

This paper concerns the evolution of a closed hypersurface of dimension in the Euclidean space under a mixed volume preserving flow. The speed equals a power of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy and Cabezas-Rivas and Sinestrari to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.

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Submitted
Oct 26, 2016
Subjects
math.DG

Comments: 33pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:1306.4539

Other curvature flowsVolume-preserving & constrained flowsCurvature pinchingHomogeneous 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)6Bertini 2016 Volume preserving non homogeneous mean curvature flow in hyperbolic space (Maria Chiara Bertini and Giuseppe Pipoli, 2016)5Bertini 2016 Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature (Heiko Kröner and Julian Scheuer, 2017)2Kröner 2017 Volume preserving flow by powers of symmetric polynomials in the principal curvatures (Maria Chiara Bertini and Carlo Sinestrari, 2017)1Bertini 2017 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)3Andrews 2018 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)8Huang 2021 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)4Wei 2022 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)7Wei 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume preserving flow by powers of symmetric polynomials in the principal curvatures

    Maria Chiara Bertini and Carlo SinestrariWider flows

    Topics Curvature pinching, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “hypersurfaces converge”, “convex hypersurfaces”, “round sphere”

  2. Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

    Heiko Kröner and Julian ScheuerWider flows

    Topics Curvature pinching, Homogeneous spaces, Long-time existence & convergence · Shared terms “pinching condition”, “curvature function”, “initial hypersurface”

  3. 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 “curvature function”, “initial hypersurface”, “convex hypersurfaces”

  4. 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”, “Gauss curvature”, “initial hypersurface”

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

    Maria Chiara Bertini and Giuseppe PipoliWider flows

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

  6. Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

    Maria Chiara Bertini and Carlo SinestrariWider flows

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

  7. 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 “Gauss curvature”, “convex hypersurfaces”

  8. 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 “initial hypersurface”, “round sphere”, “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.

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) 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. 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.