Paper · Sep 23, 2018

Curve flows with a global forcing term

Friederike Dittberner

math.DGWider flowsarXiv:1809.08643v2
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Abstract

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, 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 , and show that this condition is sharp. Secondly, for bounded forcing terms, we exclude singularities in finite time. Thirdly, for immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle.

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Submitted
Sep 23, 2018
Last revised
May 15, 2021 (v2)
Subjects
math.DG
DOI
10.1007/s12220-020-00600-1

Comments: 2 figures

Curve shorteningSingularitiesLong-time existence & convergence

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 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)6Baldauf 2018 A distance comparison principle for curve flows with a global forcing term (Friederike Dittberner, 2022)1Dittberner 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)8Bourni 2023 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)7Litzinger 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)3Bryan 2023 A curvature flow that deforms curves to an embedded target (Samuel Cuthbertson, Glen Wheeler and Valentina Wheeler, 2024)4Cuthbertson 2024 Huisken's Distance Comparison Principle in Higher Codimension (Qi Sun, 2025)2Sun 2025 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)5Hebbar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A distance comparison principle for curve flows with a global forcing term

    Friederike DittbernerWider flows

    Topics Curve shortening · Same author Friederike Dittberner · Shared terms “global forcing”, “forcing term”, “distance comparison”

  2. Huisken's Distance Comparison Principle in Higher Codimension

    Qi SunWider flows

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

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

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows

    Topics Curve shortening, Singularities · Shared terms “distance comparison”, “finite time”

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

    Samuel Cuthbertson, Glen Wheeler and Valentina WheelerWider flows

    Topics Curve shortening, Long-time existence & convergence · Shared terms “forcing term”

  5. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows

    Topics Curve shortening, Long-time existence & convergence · Shared terms “immortal”

  6. Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

    Julius Baldauf and Ao SunWider flows

    Topics Curve shortening, Singularities · Shared terms “total curvature”

  7. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening, Singularities · Shared terms “initial curve”, “finite time”

  8. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows

    Topics Curve shortening, Singularities · Shared terms “finite time”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.