Paper · Mar 10, 2021

The fundamental solutions of the curve shortening problem via the Schwarz function

Robb McDonald

math.DGWider flowsarXiv:2103.06069v3
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Abstract

Curve shortening in the -plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy , where is the Schwarz function of the curve. This equation is shown to have a parametric solution from which the known explicit solutions for curve shortening flow; the circle, grim reaper, paperclip and hairclip, can be recovered.

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Submitted
Mar 10, 2021
Last revised
Jan 14, 2022 (v3)
Subjects
math.DG, math.CV

Comments: 6 pages, 2 figures

Curve shorteningSelf-shrinkers & translators

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)7Cortissoz 2017 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)4Choi 2018 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)2Bourni 2019 The Affine Shape of a Figure-Eight under the Curve Shortening Flow (Matei P. Coiculescu and Richard Evan Schwartz, 2021)3Coiculescu 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)1Bourni 2022 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)6Zhang 2022 Generalized Elastic Translating Solitons (Alvaro Pampano, 2022)5Pampano 2022 Grim Raindrop: A Translating Solution to Curve Diffusion Flow (W. Jacob Ogden and Micah Warren, 2024)8Ogden 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

  2. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

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

    Matei P. Coiculescu and Richard Evan SchwartzWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

  4. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Curve shortening, Self-shrinkers & translators

  5. Generalized Elastic Translating Solitons

    Alvaro PampanoWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

  6. Nonconvex ancient solutions to Curve Shortening Flow

    Yongzhe Zhang, Connor Olson, Ilyas Khan et al.Wider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  8. Grim Raindrop: A Translating Solution to Curve Diffusion Flow

    W. Jacob Ogden and Micah WarrenWider flows

    Topics Curve shortening, Self-shrinkers & translators

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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.