Paper · Feb 21, 2019

Curve-shortening flow of open, elastic curves in with repelling endpoints: A minimizing movement approach

Rufat Badal

math.APWider flowsarXiv:1902.08079
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As on arXiv

Abstract

We study an -type gradient flow of an immersed elastic curve in whose endpoints repel each other via a Coulomb potential. By De Giorgi's minimizing movements scheme we prove long-time existence of the flow. The work is complemented by several numerical experiments.

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Submitted
Feb 21, 2019
Subjects
math.AP
Curve shorteningWeak & singular flowsLong-time existence & convergenceNumerics & visualization

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)4Cortissoz 2017 Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds (Harald Garcke and Robert Nürnberg, 2020)6Garcke 2020 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)8Woolgar 2021 On the H¹(ds)-gradient flow for the length functional (Philip Schrader, Glen Wheeler and Valentina-Mira Wheeler, 2021)5Schrader 2021 Generalized Elastic Translating Solitons (Alvaro Pampano, 2022)1Pampano 2022 The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature (Patrick Guidotti, 2023)2Guidotti 2023 Which shapes can appear in a Curve Shortening Flow Singularity? (Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al., 2024)3Angenent 2024 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)7Hebbar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generalized Elastic Translating Solitons

    Alvaro PampanoWider flows

    Topics Curve shortening · Shared terms “elastic curve”

  2. The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature

    Patrick GuidottiWider flows

    Topics Numerics & visualization, Curve shortening · Shared terms “numerical experiments”

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

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

    Topics Numerics & visualization, Curve shortening · Shared terms “numerical”

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  5. On the -gradient flow for the length functional

    Philip Schrader, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Numerics & visualization, Curve shortening · Shared terms “gradient flow”, “numerical”

  6. Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds

    Harald Garcke and Robert NürnbergWider flows

    Topics Numerics & visualization, Curve shortening · Shared terms “numerical”, “gradient flow”

  7. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows

    Topics Curve shortening, Long-time existence & convergence

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

    Eric Woolgar and Ran XieWider 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 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.

  3. 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.