Paper · Mar 5, 2019

Convex ancient solutions to curve shortening flow

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

math.DGWider flowsarXiv:1903.02022
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Abstract

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

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Submitted
Mar 5, 2019
Subjects
math.DG

Comments: 16 pages, 2 figures

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 2016, 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 Convex ancient solutions to mean curvature flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)4Bourni 2019 The atomic structure of ancient grain boundaries (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2020)7Bourni 2020 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)1Bourni 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)3Bourni 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)6Charyyev 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)2Bourni 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)5Bourni 2024 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)8Bourni 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 free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows

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

  2. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same author Theodora Bourni · Shared terms “convex ancient”

  3. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

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

  4. Convex ancient solutions to mean curvature flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “convex ancient”, “Wang”

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

  6. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

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

  7. The atomic structure of ancient grain boundaries

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford, Giuseppe Tinaglia · Shared terms “convex ancient”, “grim reaper”

  8. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows

    Topics Curve shortening · Same authors Theodora Bourni, Mat Langford

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.