Paper · Jun 6, 2024

Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

Yong Wei, Bo Yang, Tailong Zhou

math.DGWider flowsarXiv:2406.05159v2Pure Appl. Math. Q. Vol. 21(2025), no. 4, 1607-1643
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Abstract

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time and converges to a geodesic sphere exponentially as in the smooth topology. A key step is to show the oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence.

Comments: Final version for PAMQ. arXiv admin note: substantial text overlap with arXiv:2210.06035

Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsHomogeneous spacesLong-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 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)6Huang 2021 A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space (Yingxiang Hu, Yong Wei, Bo Yang et al., 2022)4Hu 2022 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)1Wei 2022 The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces (Shujing Pan and Bo Yang, 2023)7Pan 2023 Pinching estimates of hypersurfaces by a generalized Gauss curvature flow (Jinrong Hu and Ping Zhang, 2023)8Hu 2023 Anisotropic Gauss curvature flow of complete non-compact graphs (Shujing Pan and Yong Wei, 2024)3Pan 2024 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)5Luo 2026 Curvature measures and volume preserving flows in h-convex geometry (Xinxiu Zhang, 2026)2Zhang 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 Gauss curvature flow of convex hypersurfaces in the hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Gauss curvature flow, Volume-preserving & constrained flows, Long-time existence & convergence · Same authors Yong Wei, Bo Yang, Tailong Zhou · Shared terms “hyperbolic space”, “sphere exponentially”, “hyperbolic curvature”

  2. Curvature measures and volume preserving flows in h-convex geometry

    Xinxiu ZhangWider flows PDF

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

  3. Anisotropic Gauss curvature flow of complete non-compact graphs

    Shujing Pan and Yong WeiWider flows PDF

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “positive time”, “convex hypersurfaces”

  4. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Same authors Yong Wei, Bo Yang, Tailong Zhou · Shared terms “flow remains”, “positive time”

  5. Free boundary flows by powers of the Gauss curvature in the unit ball

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “convex hypersurfaces”

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

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows PDF

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

  7. The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

    Shujing Pan and Bo YangWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Same author Bo Yang · Shared terms “hyperbolic space”, “smooth topology”, “flow remains”

  8. Pinching estimates of hypersurfaces by a generalized Gauss curvature flow

    Jinrong Hu and Ping ZhangWider flows PDF

    Topics Gauss curvature flow, Homogeneous spaces, Other curvature flows · Shared terms “speed function”, “convex hypersurfaces”

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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) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… 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.