Paper · Apr 12, 2022

A Compact Non-convex Ancient Curve Shortening Flow

Jumageldi Charyyev

math.DGWider flowsarXiv:2204.05978
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As on arXiv

Abstract

An example of a compact, non-convex, embedded ancient solution for the curve shortening flow, which is asymptotic to Yin-Yang curve, is constructed.

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Submitted
Apr 12, 2022
Subjects
math.DG, math.AP

Comments: The same result was obtained independently by Angenent S., Khan I., Olson C., Zhang Y. in arXiv:2204.05253

Curve shorteningAncient solutions

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 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)4Bourni 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)3Bourni 2022 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)2Zhang 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)5Bourni 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)1Bourni 2024 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)7Su 2024 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)8Krznarić 2026 Length bounds and rigidity for ancient curve shortening flow on warped product surfaces (Beomseok Goh and Taehun Lee, 2026)6Goh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows

    Topics Curve shortening, Ancient solutions · Shared terms “convex ancient”

  2. Nonconvex ancient solutions to Curve Shortening Flow

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

    Topics Curve shortening, Ancient solutions · Shared terms “Yin Yang”

  3. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Curve shortening, Ancient solutions

  4. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows

    Topics Curve shortening, Ancient solutions · Shared terms “convex ancient”

  5. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows

    Topics Curve shortening, Ancient solutions · Shared terms “convex ancient”

  6. Length bounds and rigidity for ancient curve shortening flow on warped product surfaces

    Beomseok Goh and Taehun LeeWider flows

    Topics Curve shortening, Ancient solutions · Shared terms “embedded ancient”

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

    Wei-Bo Su and Kai-Wei ZhaoWider flows

    Topics Curve shortening, Ancient solutions

  8. Geometric aspects of the curve shortening flow in the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening, Ancient solutions

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.