Paper · Mar 14, 2024

Which shapes can appear in a Curve Shortening Flow Singularity?

Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene, Paige Ellingson, Ziheng Feng, Edgar Gevorgyan, Aris Lemmenes, Alex Moon, Tyler Joseph Tommasi, Yamin Zhou

math.DGWider flowsarXiv:2403.09876
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Abstract

We study possible tangles that can occur in singularities of solutions to plane Curve Shortening Flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in . As a particular example, we introduce the so-called -loop curves, which generalize Matt Grayson's Figure-Eight curve, and we conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie result, namely, a vanishing -loop, when rescaled anisotropically to fit a square bounding box, converges to a "squeezed bow-tie," i.e. the curve . As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases and on the other.

Comments: 21 pages, 14 figures

Curve shorteningSingularitiesNumerics & visualization

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 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)2Zhang 2022 Discrete hyperbolic curvature flow in the plane (Klaus Deckelnick and Robert Nürnberg, 2022)3Deckelnick 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)4Peachey 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)6Litzinger 2023 Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow (Gabriel Khan, 2023)5Khan 2023 The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature (Patrick Guidotti, 2023)1Guidotti 2023 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)7Sun 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature

    Patrick GuidottiWider flows PDF

    Topics Numerics & visualization, Curve shortening, Singularities · Shared terms “numerical”

  2. Nonconvex ancient solutions to Curve Shortening Flow

    Yongzhe Zhang, Connor Olson, Ilyas Khan et al.Wider flows PDF

    Topics Curve shortening · Same author Sigurd Angenent

  3. Discrete hyperbolic curvature flow in the plane

    Klaus Deckelnick and Robert NürnbergWider flows PDF

    Topics Numerics & visualization, Curve shortening, Singularities · Shared terms “numerical simulations”

  4. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

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

    Gabriel KhanWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “vanishing”

  6. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics Curve shortening, Singularities

  7. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows PDF

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

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

    Theodora Bourni and Martin ReirisWider flows PDF

    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.

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) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 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.

  3. 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.