Paper · Sep 1, 2023

Convex Ancient Solutions to Anisotropic Curve Shortening Flow

Theodora Bourni, Benjamin Richards

math.DGWider flowsarXiv:2309.00712
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Abstract

We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor , and a given direction and speed, this translator is unique. We then construct an ancient compact solution to anisotropic curve shortening flow, and show that this solution, along with the appropriate translating solution, are the unique solutions to anisotropic curve shortening flow that lie in a slab of a given width and no smaller.

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 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)1Bourni 2022 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)5Zhang 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)3Charyyev 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)6Bourni 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)2Bourni 2024 Grim Raindrop: A Translating Solution to Curve Diffusion Flow (W. Jacob Ogden and Micah Warren, 2024)8Ogden 2024 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)7Krznarić 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 · Same author Theodora Bourni

  2. 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 · Same author Theodora Bourni · Shared terms “convex ancient”

  3. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows PDF

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

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

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Curve shortening, Ancient solutions · Same author Theodora Bourni · Shared terms “convex ancient”

  5. Nonconvex ancient solutions to Curve Shortening Flow

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

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

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

    Theodora Bourni and Martin ReirisWider flows PDF

    Topics Curve shortening · Same author Theodora Bourni

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

    Ivan Krznarić and Rafael LópezWider flows PDF

    Topics Curve shortening, Ancient solutions

  8. Grim Raindrop: A Translating Solution to Curve Diffusion Flow

    W. Jacob Ogden and Micah WarrenWider flows PDF

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “translating solution”

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.