Paper · Jun 17, 2021

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

Matei P. Coiculescu, Richard Evan Schwartz

math.APWider flowsarXiv:2106.09213v9
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Abstract

We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case.

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Submitted
Jun 17, 2021
Last revised
Aug 23, 2022 (v9)
Subjects
math.AP, math.DG
DOI
10.4310/jdg/1721071494

Comments: Our paper was accepted to the Journal of Differential Geometry, and this version is the final version we sent to the journal. It agrees with the previous version except that we corrected a few typos and also shortened the proof of the Migration Lemma in the last chapter

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 Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case (Chengjie Yu and Feifei Zhao, 2017)6Yu 2017 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)8Choi 2018 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)4Bourni 2019 The fundamental solutions of the curve shortening problem via the Schwarz function (Robb McDonald, 2021)5McDonald 2021 Some New Results in Geometric Analysis (Matei P. Coiculescu, 2021)1Coiculescu 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)2Bourni 2022 Generalized Elastic Translating Solitons (Alvaro Pampano, 2022)7Pampano 2022 Which shapes can appear in a Curve Shortening Flow Singularity? (Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al., 2024)3Angenent 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some New Results in Geometric Analysis

    Matei P. CoiculescuWider flows

    Topics Curve shortening · Same author Matei P. Coiculescu · Shared terms “figure eight”, “eight curves”

  2. 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”

  3. Which shapes can appear in a Curve Shortening Flow Singularity?

    Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al.Wider flows

    Topics Curve shortening · Shared terms “figure eight”, “bounding box”, “eight curves”

  4. 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”, “Angenent”

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

    Robb McDonaldWider flows

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

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

    Chengjie Yu and Feifei ZhaoWider flows

    Topics Curve shortening · Shared terms “transformations”

  7. Generalized Elastic Translating Solitons

    Alvaro PampanoWider flows

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

  8. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider 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.