Paper · May 26, 2022

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

Friederike Dittberner

math.DGWider flowsarXiv:2205.13466
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider closed, embedded, smooth curves in the plane and study their behaviour under curve flows with a global forcing term. We prove an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below -pi and arbitrary global forcing terms.

View PDF View HTML arXiv page
Submitted
May 26, 2022
Subjects
math.DG

Comments: 5 pages

Curve shortening

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 Stabilization technique applied to curve shortening flow in ℝ³ (Hayk Mikayelyan, 2022)7Mikayelyan 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)4Peachey 2022 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)3Langford 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)2Bryan 2023 The continuity of curvature flow in Fréchet distance (Shiyi Ma, 2023)6Ma 2023 Delayed parabolic regularity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 A curvature flow that deforms curves to an embedded target (Samuel Cuthbertson, Glen Wheeler and Valentina Wheeler, 2024)5Cuthbertson 2024 Huisken's Distance Comparison Principle in Higher Codimension (Qi Sun, 2025)1Sun 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Huisken's Distance Comparison Principle in Higher Codimension

    Qi SunWider flows

    Topics Curve shortening · Shared terms “distance comparison”, “comparison principle”, “Huisken distance”

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

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows

    Topics Curve shortening · Shared terms “distance comparison”

  3. A distance comparison principle for curve shortening flow with free boundary

    Mat Langford and Jonathan J. ZhuWider flows

    Topics Curve shortening · Shared terms “distance comparison”, “comparison principle”

  4. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  5. A curvature flow that deforms curves to an embedded target

    Samuel Cuthbertson, Glen Wheeler and Valentina WheelerWider flows

    Topics Curve shortening · Shared terms “forcing term”

  6. The continuity of curvature flow in Fréchet distance

    Shiyi MaWider flows

    Topics Curve shortening · Shared terms “initial curves”

  7. Stabilization technique applied to curve shortening flow in

    Hayk MikayelyanWider flows

    Topics Curve shortening · Shared terms “Huisken”

  8. Delayed parabolic regularity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider 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)

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.