Paper · Jan 21, 2016

The evolution of Jordan curves on by curve shortening flow

Joseph Lauer

math.APWider flowsarXiv:1601.05704
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Abstract

In this paper we prove that if is a Jordan curve on then there is a smooth curve shortening flow defined on which converges to in as . Another perspective is that the level-set flow of is smooth. This is a generalization of the author's previous work where the planar case was studied. If a Jordan curve on has Lebesgue measure zero then we show that the level-set flow instantly becomes a smooth closed curve. If the Lebesgue measure is positive then for small time the level-set flow is an annulus with smooth boundary. This second case should be interpreted as a failure of uniqueness. As in the planar case a key step in the proof is establishing a length estimate for smooth curves that depends on a geometric quantity called the -multiplicity. The majority of this paper concerns the extension of this length estimate to .

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Submitted
Jan 21, 2016
Subjects
math.AP, math.DG, math.GN

Comments: 21 pages, 4 figures

Curve shorteningWeak & singular flowsStability & uniqueness

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 Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case (Chengjie Yu and Feifei Zhao, 2017)8Yu 2017 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)7Woolgar 2021 Continuous dependence of curvature flow on initial conditions (Michael Gene Dobbins, 2021)1Dobbins 2021 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 The continuity of curvature flow in Fréchet distance (Shiyi Ma, 2023)2Ma 2023 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)6Sobnack 2024 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)5Sun 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Continuous dependence of curvature flow on initial conditions

    Michael Gene DobbinsWider flows

    Topics Curve shortening, Weak & singular flows, Stability & uniqueness · Shared terms “Jordan curve”, “set flow”, “level set”

  2. The continuity of curvature flow in Fréchet distance

    Shiyi MaWider flows

    Topics Curve shortening, Weak & singular flows · Shared terms “Jordan curve”, “set flow”, “level set”

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening, Stability & uniqueness

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  5. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows

    Topics Curve shortening, Stability & uniqueness · Shared terms “smooth closed”

  6. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening · Shared terms “closed curves”

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

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

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

    Chengjie Yu and Feifei ZhaoWider 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.