Paper · Dec 15, 2023

The continuity of curvature flow in Fréchet distance

Shiyi Ma

math.DGWider flowsarXiv:2312.15005
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Abstract

We study the evolution of a Jordan curve on the plane by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Fréchet distance.

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Submitted
Dec 15, 2023
Subjects
math.DG, math.AP, math.MG

Comments: 11 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2106.08907 by other authors

Curve shorteningOther curvature flowsWeak & singular flows

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 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)1Peachey 2022 A distance comparison principle for curve flows with a global forcing term (Friederike Dittberner, 2022)3Dittberner 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)5Litzinger 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)4Bryan 2023 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)6Sobnack 2024 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)7Sun 2024 Constructing λ-Angenent curve by flow method (Pak Tung Ho, 2026)8Ho 2026 The inverse curve shortening flow on the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)2Krznarić 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  2. The inverse curve shortening flow on the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening

  3. A distance comparison principle for curve flows with a global forcing term

    Friederike DittbernerWider flows

    Topics Curve shortening · Shared terms “initial curves”

  4. Sharp distance comparison for curve shortening flow on the round sphere

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows

    Topics Curve shortening

  5. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening · Shared terms “initial curves”

  6. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening

  7. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows

    Topics Curve shortening

  8. Constructing -Angenent curve by flow method

    Pak Tung HoWider 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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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)

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.