Paper · Apr 8, 2024

Deforming Locally Convex Curves into Curves of Constant -order Width

Laiyuan Gao, Horst Martini, Deyan Zhang

math.DGWider flowsarXiv:2404.05267
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Abstract

A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant -order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is -symmetric.

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Submitted
Apr 8, 2024
Subjects
math.DG
Other curvature flowsAnisotropic & nonlocal flowsLong-time existence & convergence

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

    Ze-Yu Ye and Xiao-Liu WangWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Shared terms “locally convex”, “convex curve”, “evolving curves”

  2. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows

  3. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  4. The nonlocal mean curvature flow of periodic graphs

    Bogdan-Vasile Matioc and Christoph WalkerWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence

  5. Variational Nonlinear and Nonlocal Curvature Flows

    Daniele De GennaroWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  6. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  7. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  8. Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

    De Gennaro Daniele, Andrea Kubin and Anna KubinWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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)