Paper · Aug 31, 2017

Quermassintegral preserving curvature flow in Hyperbolic space

Ben Andrews, Yong Wei

math.DGWider flowsarXiv:1708.09583v2
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Abstract

We consider the quermassintegral preserving flow of closed h-convex hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function of the principal curvatures which is inverse concave and has dual approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is h-convex, then the solution of the flow becomes strictly h-convex for , the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

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Submitted
Aug 31, 2017
Last revised
May 17, 2018 (v2)
Subjects
math.DG, math.AP
Journal
Geometric and Functional Analysis, Vol 28 (2018), Issue 5, pp 1183-1208
DOI
10.1007/s00039-018-0456-9

Comments: 22 pages, minor revision, accepted by GAFA

Other curvature flowsHomogeneous spacesLong-time existence & convergenceGeometric inequalities

Its neighbourhood

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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 New pinching estimates for Inverse curvature flows in space forms (Yong Wei, 2017)7Wei 2017 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)1Andrews 2018 Locally constrained curvature flows and geometric inequalities in hyperbolic space (Yingxiang Hu, Haizhong Li and Yong Wei, 2020)6Hu 2020 Shifted inverse curvature flows in hyperbolic space (Xianfeng Wang, Yong Wei and Tailong Zhou, 2020)8Wang 2020 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)3Wei 2022 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)4Wei 2024 The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere (Sara Albert-Niclòs, Esther Cabezas-Rivas and Shujing Pan, 2026)5Albert-Niclò 2026 Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space (Tianci Luo, Yong Wei, Ruijia Zhang et al., 2026)2Luo 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 flow and Alexandrov-Fenchel type inequalities in hyperbolic space

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Geometric inequalities, Homogeneous spaces, Long-time existence & convergence · Same authors Ben Andrews, Yong Wei · Shared terms “hyperbolic space”, “sphere exponentially”, “preserving flow”

  2. Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows

    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “quermassintegral preserving”, “hyperbolic space”, “preserving curvature”

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

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “preserving curvature”, “sphere exponentially”

  4. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “sphere exponentially”, “smooth topology”

  5. The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

    Sara Albert-Niclòs, Esther Cabezas-Rivas and Shujing PanWider flows

    Topics Geometric inequalities, Homogeneous spaces, Long-time existence & convergence · Shared terms “quermassintegral preserving”, “preserving curvature”, “convex hypersurfaces”

  6. Locally constrained curvature flows and geometric inequalities in hyperbolic space

    Yingxiang Hu, Haizhong Li and Yong WeiWider flows

    Topics Geometric inequalities, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “quermassintegral”, “convex hypersurfaces”

  7. New pinching estimates for Inverse curvature flows in space forms

    Yong WeiWider flows

    Topics Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “approaching zero”, “positive cone”, “inverse concave”

  8. Shifted inverse curvature flows in hyperbolic space

    Xianfeng Wang, Yong Wei and Tailong ZhouWider flows

    Topics Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “strictly increasing”, “positive power”

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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.