Paper · Sep 8, 2017

New pinching estimates for Inverse curvature flows in space forms

Yong Wei

math.DGWider flowsarXiv:1709.02546v2
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Abstract

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form of constant sectional curvature with speed given by , where for and for , is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual approaching zero on the boundary of the positive cone . We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flow.

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Submitted
Sep 8, 2017
Last revised
Apr 1, 2018 (v2)
Subjects
math.DG, math.AP
Journal
The Journal of Geometric Analysis, Vol 29, Issue 2, pp 1555-1570, 2019
DOI
10.1007/s12220-018-0051-1

Comments: 15 pages,v2

Other curvature flowsInverse mean curvature flowCurvature pinchingSphere theoremsHomogeneous spaces

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 Quermassintegral preserving curvature flow in Hyperbolic space (Ben Andrews and Yong Wei, 2017)3Andrews 2017 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)7Andrews 2018 A Pinching Estimate for Convex Hypersurfaces Evolving Under a Nonhomogeneous Variant of Mean Curvature Flow (Tim Espin, 2020)5Espin 2020 Shifted inverse curvature flows in hyperbolic space (Xianfeng Wang, Yong Wei and Tailong Zhou, 2020)2Wang 2020 On an inverse curvature flow in two-dimensional space forms (Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al., 2021)1Kwong 2021 Pinching estimates of hypersurfaces by a generalized Gauss curvature flow (Jinrong Hu and Ping Zhang, 2023)8Hu 2023 Contraction of hypersurfaces with positive sectional curvature in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)4Luo 2026 Entropy and round-point convergence for the σₖ-curvature flow (Tianci Luo, Yong Wei, Ruijia Zhang et al., 2026)6Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On an inverse curvature flow in two-dimensional space forms

    Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al.Wider flows

    Topics Sphere theorems, Inverse mean curvature flow, Other curvature flows · Same author Yong Wei · Shared terms “space form”

  2. Shifted inverse curvature flows in hyperbolic space

    Xianfeng Wang, Yong Wei and Tailong ZhouWider flows

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “principal curvatures”, “convex hypersurfaces”

  3. Quermassintegral preserving curvature flow in Hyperbolic space

    Ben Andrews and Yong WeiWider flows

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

  4. Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

    Tianci Luo, Yong Wei and Rong ZhouWider flows

    Topics Sphere theorems, Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “sectional curvature”, “principal curvatures”

  5. A Pinching Estimate for Convex Hypersurfaces Evolving Under a Nonhomogeneous Variant of Mean Curvature Flow

    Tim EspinWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “pinching estimate”, “convex hypersurfaces”, “principal curvatures”

  6. Entropy and round-point convergence for the -curvature flow

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

    Topics Curvature pinching, Other curvature flows · Same author Yong Wei · Shared terms “strictly convex”, “convex hypersurfaces”

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

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Sphere theorems, Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “principal curvatures”, “convex hypersurfaces”, “sectional curvature”

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

    Jinrong Hu and Ping ZhangWider flows

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “pinching estimate”, “convex hypersurfaces”

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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

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.