Paper · Jul 29, 2018

Convergence of Curve Shortening Flow to Translating Soliton

Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos

math.DGWider flowsarXiv:1807.11100
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Abstract

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in under the -curve shortening flow for exponents . We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under -curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case , and we prove for all exponents up to the critical case .

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Submitted
Jul 29, 2018
Subjects
math.DG

Comments: 28 pages, 0 figure

Curve shorteningSelf-shrinkers & translatorsBoundaries & non-compact

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 Translating solutions to the Gauss curvature flow with flat sides (Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm Lee, 2016)7Choi 2016 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)4Cortissoz 2017 Convergence of Gauss curvature flows to translating solitons (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2019)1Choi 2019 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)8Bourni 2019 Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2020)6Choi 2020 Generalized Elastic Translating Solitons (Alvaro Pampano, 2022)3Pampano 2022 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)5Choi 2024 Classification of ancient finite-entropy curve shortening flows (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2026)2Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of Gauss curvature flows to translating solitons

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same authors Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “translating solitons”, “compact convex”, “asymptotic behavior”

  2. Classification of ancient finite-entropy curve shortening flows

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows

    Topics Curve shortening, Self-shrinkers & translators, Boundaries & non-compact · Same author Kyeongsu Choi

  3. Generalized Elastic Translating Solitons

    Alvaro PampanoWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “translating solitons”

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  5. Uniqueness of tangent flows at infinity for finite-entropy shortening curves

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows

    Topics Curve shortening, Self-shrinkers & translators · Same author Kyeongsu Choi · Shared terms “addition”

  6. Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same authors Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “translating solitons”

  7. Translating solutions to the Gauss curvature flow with flat sides

    Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm LeeWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same authors Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “translating solitons”

  8. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.