Paper · May 11, 2023

Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

Gabriel Khan

math.DGWider flowsarXiv:2305.07171Bull. Aust. Math. Soc. 109 (2024) 571-580
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Abstract

We study curve-shortening flow for twisted curves in (i.e., curves with nowhere vanishing curvature and torsion ) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type II singularity which admits sequences along which . This contrasts strongly with Altschuler's planarity theorem [J. Differential Geom. (1991)], which shows that along any essential blow-up sequence, .

Comments: 10 pages

Curve shorteningSingularities

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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)2Peachey 2022 Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)5Minarčík 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)4Bourni 2023 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)1Litzinger 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)3Bryan 2023 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)7Cuthbertson 2024 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)6Sun 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 low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “Altschuler”

  2. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

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

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “Geom”

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

    Theodora Bourni and Martin ReirisWider flows PDF

    Topics Curve shortening, Singularities

  5. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “contrast”

  6. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows PDF

    Topics Curve shortening, Singularities

  7. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows PDF

    Topics Curve shortening · Shared terms “eventual”

  8. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    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) 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.