Paper · Jan 19, 2021

Self-similar curve shortening flow in hyperbolic 2-space

Eric Woolgar, Ran Xie

math.DGWider flowsarXiv:2101.07469v2
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Abstract

We find and classify self-similar solutions of the curve shortening flow in standard hyperbolic 2-space. Together with earlier work of Halldórsson on curve shortening flow in the plane and Santos dos Reis and Tenenblat in the 2-sphere, this completes the classification of self-similar curve shortening flows in the constant curvature model spaces in 2-dimensions.

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Submitted
Jan 19, 2021
Last revised
Feb 21, 2021 (v2)
Subjects
math.DG
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 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)1Cortissoz 2017 From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow (Yann Brenier and Xianglong Duan, 2017)6Brenier 2017 Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case (Chengjie Yu and Feifei Zhao, 2017)2Yu 2017 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)8Choi 2018 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)7Ma 2020 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)5Charyyev 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)4Krznarić 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

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

    Chengjie Yu and Feifei ZhaoWider flows

    Topics Curve shortening

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  4. Geometric aspects of the curve shortening flow in the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening

  5. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

    Topics Curve shortening

  6. From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow

    Yann Brenier and Xianglong DuanWider flows

    Topics Curve shortening

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

    John Man Shun MaWider flows

    Topics Curve shortening

  8. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider 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.