Paper · Apr 5, 2023

Singularities of low entropy high codimension curve shortening flow

Florian Litzinger

math.DGWider flowsarXiv:2304.02487
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Abstract

We consider curve shortening flow of arbitrary codimension in an Euclidean background. We show that, close to a singularity, the flow is asymptotically planar, paralleling Altschuler's work in the case of space curves, and analyse the blow-up limits of the flow. Using these results, we then prove that the curve shortening flow of initial curves with an entropy bound converges to a round point in finite time.

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Submitted
Apr 5, 2023
Subjects
math.DG, math.AP

Comments: 20 pages

Curve 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 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)6Peachey 2022 Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)3Minarčík 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)7Bourni 2023 Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow (Gabriel Khan, 2023)2Khan 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)4Bryan 2023 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)8Sun 2024 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)5Sun 2025 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)1Nguyen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. High Codimension Curve Shortening Flow with Free Boundary

    Huy The Nguyen and Artemis A. VogiatziWider flows

    Topics Curve shortening, Singularities · Shared terms “codimension curve”, “high codimension”, “low entropy”

  2. Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

    Gabriel KhanWider flows

    Topics Curve shortening, Singularities · Shared terms “Altschuler”

  3. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows

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

  5. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows

    Topics Curve shortening, Singularities · Shared terms “round point”

  6. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

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

    Theodora Bourni and Martin ReirisWider flows

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

  8. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows

    Topics Curve shortening · Shared terms “space curves”

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 usually cannot run forever: curvature blows up in finite time, 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.