Paper · Mar 7, 2021

On an inverse curvature flow in two-dimensional space forms

Kwok-Kun Kwong, Yong Wei, Glen Wheeler, Valentina-Mira Wheeler

math.DGWider flowsarXiv:2103.04338
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As on arXiv

Abstract

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetricinequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the case of a conjecture of Girão-Pinheiro.

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Submitted
Mar 7, 2021
Subjects
math.DG, math.AP
Journal
Mathematische Annalen, (2022) 384: 285-308
DOI
10.1007/s00208-021-02285-5

Comments: 17 pages

Other curvature flowsInverse mean curvature flowSurfacesSphere theoremsLong-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 Quermassintegral preserving curvature flow in Hyperbolic space (Ben Andrews and Yong Wei, 2017)6Andrews 2017 New pinching estimates for Inverse curvature flows in space forms (Yong Wei, 2017)2Wei 2017 On a curvature flow model for embryonic epidermal wound healing (Shuhui He, Glen Wheeler and Valentina-Mira Wheeler, 2018)7He 2018 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)4Andrews 2018 Shifted inverse curvature flows in hyperbolic space (Xianfeng Wang, Yong Wei and Tailong Zhou, 2020)5Wang 2020 Minkowski inequalities and constrained inverse curvature flows in warped spaces (Julian Scheuer, 2020)8Scheuer 2020 Upper bounds for the Alexandrov-Fenchel deficit via integral formulas (Kwok-Kun Kwong and Yong Wei, 2025)1Kwong 2025 Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space (Tianci Luo, Yong Wei, Ruijia Zhang et al., 2026)3Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

    Kwok-Kun Kwong and Yong WeiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same authors Kwok-Kun Kwong, Yong Wei · Shared terms “weighted geometric”, “support function”

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

    Yong WeiWider flows

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

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

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

    Topics Geometric inequalities, Sphere theorems, Long-time existence & convergence · Same author Yong Wei · Shared terms “counterexamples”

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

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Geometric inequalities, Sphere theorems, Long-time existence & convergence · Same author Yong Wei · Shared terms “smooth topology”

  5. Shifted inverse curvature flows in hyperbolic space

    Xianfeng Wang, Yong Wei and Tailong ZhouWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Yong Wei · Shared terms “counterexamples”

  6. Quermassintegral preserving curvature flow in Hyperbolic space

    Ben Andrews and Yong WeiWider flows

    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “smooth topology”

  7. On a curvature flow model for embryonic epidermal wound healing

    Shuhui He, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Other curvature flows · Same authors Glen Wheeler, Valentina-Mira Wheeler · Shared terms “convex curve”

  8. Minkowski inequalities and constrained inverse curvature flows in warped spaces

    Julian ScheuerWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Surfaces

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.