Paper · May 27, 2023

Two nonlocal inverse curvature flows of convex closed plane curves

Zezhen Sun

math.DGWider flowsarXiv:2305.17314International Journal of Mathematics.Vol. 35, No. 2 (2024) 2350107
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Abstract

In this paper we introduce two -type () curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. And finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

Comments: 14 pages. arXiv admin note: text overlap with arXiv:1408.1805 by other authors

Other curvature flowsInverse mean curvature flowSingularitiesLong-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 On potentials whose level sets are orbits (Philippe Bolle, Marco Mazzucchelli and Andrea Venturelli, 2024)7Bolle 2024 On length-preserving and area-preserving inverse curvature flow of planar curves with singularities (Yunlong Yang, Yanwen Zhao, Jianbo Fang et al., 2024)5Yang 2024 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)6Gao 2024 On a non-local area-preserving curve flow (Zezhen Sun and Yuting Wu, 2025)2Sun 2025 On an area-preserving inverse curvature flow for plane curves (Zezhen Sun and Yuting Wu, 2025)1Sun 2025 Inverse curvature flow of closed Legendre curves (Takashi Kagaya and Masatomo Takahashi, 2025)4Kagaya 2025 On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane (Zhishuai Liu and Guoxin Wei, 2026)3Liu 2026 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)8Liu 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 area-preserving inverse curvature flow for plane curves

    Zezhen Sun and Yuting WuWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same author Zezhen Sun · Shared terms “plane curves”, “closed plane”, “convex closed”

  2. On a non-local area-preserving curve flow

    Zezhen Sun and Yuting WuWider flows PDF

    Topics Other curvature flows · Same author Zezhen Sun · Shared terms “finite circle”, “closed convex”, “evolution process”

  3. On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “curves converge”, “evolving curves”, “converges smoothly”

  4. Inverse curvature flow of closed Legendre curves

    Takashi Kagaya and Masatomo TakahashiWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  5. On length-preserving and area-preserving inverse curvature flow of planar curves with singularities

    Yunlong Yang, Yanwen Zhao, Jianbo Fang et al.Wider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows, Singularities · Shared terms “planar curves”, “nonlocal”

  6. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows PDF

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “converges smoothly”

  7. On potentials whose level sets are orbits

    Philippe Bolle, Marco Mazzucchelli and Andrea VenturelliWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “increasing”

  8. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “closed plane”, “convex closed”, “plane curves”

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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) 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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