Paper · Dec 3, 2020

Classification of ancient flows by sub-affine-critical powers of curvature in

Kyeongsu Choi, Liming Sun

math.DGarXiv:2012.01727v2
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Abstract

We classify closed convex -curve shortening flows for sub-affine-critical powers . In addition, we show that closed convex smooth finite entropy -curve shortening flows with is a shrinking circle. After normalization, the ancient flows satisfying the above conditions converge exponentially fast to smooth closed convex shrinkers at the backward infinity. In particular, when with , the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.

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Submitted
Dec 3, 2020
Last revised
Feb 2, 2022 (v2)
Subjects
math.DG, math.AP
Curve shorteningRicci solitonsAncient solutionsLong-time existence & convergence

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 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)4Choi 2018 Ancient gradient flows of elliptic functionals and Morse index (Kyeongsu Choi and Christos Mantoulidis, 2019)3Choi 2019 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)5Bourni 2019 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)7Ma 2020 Mean curvature flow with generic initial data (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2020)8Chodosh 2020 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)6Charyyev 2022 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)2Choi 2024 Classification of ancient finite-entropy curve shortening flows (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2026)1Choi 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 ancient finite-entropy curve shortening flows

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

    Topics Curve shortening, Ancient solutions · Same author Kyeongsu Choi · Shared terms “finite entropy”, “shrinking circle”

  2. Uniqueness of tangent flows at infinity for finite-entropy shortening curves

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

    Topics Curve shortening, Ancient solutions, Long-time existence & convergence · Same author Kyeongsu Choi · Shared terms “finite entropy”, “shrinking circle”, “exponentially fast”

  3. Ancient gradient flows of elliptic functionals and Morse index

    Kyeongsu Choi and Christos MantoulidisWider flows

    Topics Curve shortening, Ancient solutions · Same author Kyeongsu Choi · Shared terms “shrinking circle”

  4. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Curve shortening · Same author Kyeongsu Choi · Shared terms “addition”

  5. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Ancient solutions · Shared terms “shrinking circle”

  6. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

    Topics Curve shortening, Ancient solutions

  7. Ancient solutions to the Curve Shortening Flow spanning the halfplane

    John Man Shun MaWider flows

    Topics Curve shortening, Ancient solutions

  8. Mean curvature flow with generic initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.

    Topics Ancient solutions, Long-time existence & convergence, Ricci solitons · Same author Kyeongsu Choi

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 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.

  3. 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.