Paper · Feb 22, 2017

On the rate of convergence of the -curve shortening flow

Jean C. Cortissoz, Andrés Galindo, Alexander Murcia

math.APWider flowsarXiv:1702.06983
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Abstract

In this paper we give rates of convergence for the -curve shortening flow for an integer, which improves on the known estimates and which are probably sharp.

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Submitted
Feb 22, 2017
Subjects
math.AP

Comments: Any, comments and suggestions are more than welcome

Curve shortening

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 Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case (Chengjie Yu and Feifei Zhao, 2017)4Yu 2017 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)3Choi 2018 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)6Ma 2020 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)1Woolgar 2021 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)5Charyyev 2022 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)7dos Reis 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)2Peachey 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)8Bourni 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Self-similar curve shortening flow in hyperbolic 2-space

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  2. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  3. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Curve shortening

  4. Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case

    Chengjie Yu and Feifei ZhaoWider flows

    Topics Curve shortening

  5. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

    Topics Curve shortening

  6. Ancient solutions to the Curve Shortening Flow spanning the halfplane

    John Man Shun MaWider flows

    Topics Curve shortening

  7. Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Topics Curve shortening

  8. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows

    Topics Curve shortening

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.

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.