Paper · Apr 11, 2022

Nonconvex ancient solutions to Curve Shortening Flow

Yongzhe Zhang, Connor Olson, Ilyas Khan, Sigurd Angenent

math.DGWider flowsarXiv:2204.05253v3
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Abstract

We construct an ancient solution to planar curve shortening. The solution is at all times compact and embedded. For it is approximated by the rotating Yin-Yang soliton, truncated at a finite angle , and closed off by a small copy of the Grim Reaper translating soliton.

Comments: The same result was independently obtained by Charyyev, J. in arXiv:2204.05978

Curve shorteningAncient solutionsSelf-shrinkers & translators

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 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)1Bourni 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)2Charyyev 2022 Generalized Elastic Translating Solitons (Alvaro Pampano, 2022)6Pampano 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)3Bourni 2023 Ancient curve shortening flow in the disc with mixed boundary condition (Mat Langford, Yuxing Liu and George McNamara, 2024)7Langford 2024 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)8Bourni 2024 Classification of ancient finite-entropy curve shortening flows (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2026)4Choi 2026 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)5Krznarić 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows PDF

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Shared terms “grim reaper”

  2. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows PDF

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

  3. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows PDF

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators

  4. Classification of ancient finite-entropy curve shortening flows

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows PDF

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Shared terms “grim reaper”

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

    Ivan Krznarić and Rafael LópezWider flows PDF

    Topics Curve shortening, Ancient solutions

  6. Generalized Elastic Translating Solitons

    Alvaro PampanoWider flows PDF

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “translating solitons”, “grim reaper”

  7. Ancient curve shortening flow in the disc with mixed boundary condition

    Mat Langford, Yuxing Liu and George McNamaraWider flows PDF

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Shared terms “grim reaper”

  8. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows PDF

    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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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.