Paper · Nov 24, 2017

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

Chengjie Yu, Feifei Zhao

math.DGWider flowsarXiv:1711.08872
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Abstract

In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give descriptions of the solutions for the degenerate case. Some new special solutions for the affine curve shortening flow are found.

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Submitted
Nov 24, 2017
Subjects
math.DG

Comments: 27 pages. All comments are 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 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)2Cortissoz 2017 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)7Ma 2020 The curve shortening flow in the metric-affine plane (Vladimir Rovenski, 2020)1Rovenski 2020 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)3Woolgar 2021 The Affine Shape of a Figure-Eight under the Curve Shortening Flow (Matei P. Coiculescu and Richard Evan Schwartz, 2021)5Coiculescu 2021 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)8Charyyev 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)4Peachey 2022 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)6Krznarić 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The curve shortening flow in the metric-affine plane

    Vladimir RovenskiWider flows

    Topics Curve shortening · Shared terms “Euclidean plane”

  2. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  3. Self-similar curve shortening flow in hyperbolic 2-space

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  4. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  5. The Affine Shape of a Figure-Eight under the Curve Shortening Flow

    Matei P. Coiculescu and Richard Evan SchwartzWider flows

    Topics Curve shortening · Shared terms “transformations”

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

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening

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

    John Man Shun MaWider flows

    Topics Curve shortening

  8. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider 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.