Paper · Aug 12, 2018

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

Julius Baldauf, Ao Sun

math.DGWider flowsarXiv:1808.03936v2
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Abstract

We prove that a closed immersed plane curve with total curvature has entropy at least times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature whose entropy is less than times the entropy of the embedded circle. As an application, we extend Colding-Minicozzi's notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.

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Submitted
Aug 12, 2018
Last revised
Dec 25, 2020 (v2)
Subjects
math.DG

Comments: 22 pages, final version. To appear in Communications in Analysis and Geometry

Mean curvature flowCurve shorteningSingularities

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 Type I singularities in the curve shortening flow associated to a density (Vicente Miquel and Francisco Viñado-Lereu, 2016)8Miquel 2016 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)6Cortissoz 2017 Construction of High Codimension Ancient Mean Curvature Flows (Douglas Stryker and Ao Sun, 2019)5Stryker 2019 Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at -∞ (Douglas Stryker and Ao Sun, 2019)3Stryker 2019 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)4Litzinger 2023 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)2Su 2024 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)7Sun 2024 Singularities of Curve Shortening Flow with Convex Projections (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. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows

    Topics Curve shortening, Singularities · Shared terms “closed immersed”

  2. On bounds of entropy and total curvature for ancient curve shortening flows

    Wei-Bo Su and Kai-Wei ZhaoWider flows

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

  3. Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at

    Douglas Stryker and Ao SunWider flows

    Topics Curve shortening, Singularities · Same author Ao Sun

  4. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening, Singularities · Shared terms “entropy bound”

  5. Construction of High Codimension Ancient Mean Curvature Flows

    Douglas Stryker and Ao SunWider flows

    Topics Curve shortening · Same author Ao Sun

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  7. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows

    Topics Curve shortening · Shared terms “closed immersed”

  8. Type I singularities in the curve shortening flow associated to a density

    Vicente Miquel and Francisco Viñado-LereuWider flows

    Topics Curve shortening, Singularities

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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)

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.