Paper · Nov 8, 2022

Generalized Elastic Translating Solitons

Alvaro Pampano

math.DGWider flowsarXiv:2211.04603
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Abstract

We study translating soliton solutions to the flow by powers of the curvature of curves in the plane. We characterize these solitons as critical curves for functionals depending on the curvature. More precisely, translating solitons to the flow by powers of the curvature are shown to be generalized elastic curves. In particular, focusing on the curve shortening flow, we deduce a new variational characterization of the grim reaper curve.

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Submitted
Nov 8, 2022
Subjects
math.DG
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 2021, 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 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)3Zhang 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)7Peachey 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)5Bourni 2023 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)6Choi 2024 Grim Raindrop: A Translating Solution to Curve Diffusion Flow (W. Jacob Ogden and Micah Warren, 2024)2Ogden 2024 Classification of ancient finite-entropy curve shortening flows (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2026)4Choi 2026 The inverse curve shortening flow on the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)8Krznarić 2026
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. Grim Raindrop: A Translating Solution to Curve Diffusion Flow

    W. Jacob Ogden and Micah WarrenWider flows

    Topics Curve shortening, Self-shrinkers & translators

  3. 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 “translating solitons”, “grim reaper”

  4. 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 · Shared terms “reaper curves”, “grim reaper”

  5. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows

    Topics Curve shortening, Self-shrinkers & translators

  6. 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 · Shared terms “reaper curves”, “grim reaper”

  7. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  8. The inverse curve shortening flow on the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider 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) 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.