Paper · Jul 14, 2020

Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

Laiyuan Gao

math.DGWider flowsarXiv:2007.07342
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Abstract

Let be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve into . It is proved that this flow exits globally, preserves both the local convexity and the elastic energy of the evolving curve. If the two curves have same elastic energy then the curvature flow deforms the evolving curve into the target curve as time tends to infinity.

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Submitted
Jul 14, 2020
Subjects
math.DG

Comments: 15 pages, 2 figures

Other curvature flows

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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 Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows (Natasa Sesum, Dong-Ho Tsai and Xiao-Liu Wang, 2017)3Sesum 2017 A curvature flow in the plane with a nonlocal term (Luis Caffarelli and Hui Yu, 2017)6Caffarelli 2017 Two nonlocal inverse curvature flows of convex closed plane curves (Zezhen Sun, 2023)7Sun 2023 Deforming Locally Convex Curves into Curves of Constant k-order Width (Laiyuan Gao, Horst Martini and Deyan Zhang, 2024)1Gao 2024 A curvature flow that deforms curves to an embedded target (Samuel Cuthbertson, Glen Wheeler and Valentina Wheeler, 2024)4Cuthbertson 2024 Star-shaped Curves under Gage's Area-preserving Flow and the CSF (Laiyuan Gao, Shicheng Zhang and Yuntao Zhang, 2024)2Gao 2024 Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions (Giovanni Bellettini, Shokhrukh Yu. Kholmatov and Matteo Novaga, 2025)8Bellettini 2025 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 Wang, 2026)5Ye 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Deforming Locally Convex Curves into Curves of Constant -order Width

    Laiyuan Gao, Horst Martini and Deyan ZhangWider flows

    Topics Other curvature flows · Same author Laiyuan Gao · Shared terms “locally convex”, “convex curve”, “elastic energy”

  2. Star-shaped Curves under Gage's Area-preserving Flow and the CSF

    Laiyuan Gao, Shicheng Zhang and Yuntao ZhangWider flows

    Same author Laiyuan Gao · Shared terms “evolving curves”, “globally”, “preserves”

  3. Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows

    Natasa Sesum, Dong-Ho Tsai and Xiao-Liu WangWider flows

    Topics Other curvature flows · Shared terms “locally convex”, “nonlocal”

  4. A curvature flow that deforms curves to an embedded target

    Samuel Cuthbertson, Glen Wheeler and Valentina WheelerWider flows

    Topics Other curvature flows · Shared terms “target curve”

  5. 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 Other curvature flows · Shared terms “locally convex”, “convex curve”, “evolving curves”

  6. A curvature flow in the plane with a nonlocal term

    Luis Caffarelli and Hui YuWider flows

    Topics Other curvature flows · Shared terms “nonlocal term”

  7. Two nonlocal inverse curvature flows of convex closed plane curves

    Zezhen SunWider flows

    Topics Other curvature flows · Shared terms “evolving curves”, “nonlocal”

  8. Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions

    Giovanni Bellettini, Shokhrukh Yu. Kholmatov and Matteo NovagaWider flows

    Topics Other curvature flows · Shared terms “time tends”, “globally”, “preserves”

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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)