Paper · Aug 5, 2026

Non-uniqueness of geodesic limits and a question of Grayson and Gage

Shrey Aryan, Tang-Kai Lee

math.DGWider flowsarXiv:2608.04855
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Abstract

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

Comments: 32 Pages

Curve shorteningStability & uniquenessLong-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 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 Curve shortening flows on rotational surfaces generated by monotone convex functions (Naotoshi Fujihara, 2022)6Fujihara 2022 Surfaces of section for geodesic flows of closed surfaces (Gonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli et al., 2022)4Contreras 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 From curve shortening to flat link stability and Birkhoff sections of geodesic flows (Marcelo R. R. Alves and Marco Mazzucchelli, 2024)2Alves 2024 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)7Sobnack 2024 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)5Fujihara 2024 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)8Sun 2025 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)1Hebbar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows PDF

    Topics Curve shortening, Long-time existence & convergence · Shared terms “immortal curve”, “closed geodesic”

  2. From curve shortening to flat link stability and Birkhoff sections of geodesic flows

    Marcelo R. R. Alves and Marco MazzucchelliWider flows PDF

    Topics Curve shortening, Stability & uniqueness · Shared terms “simple closed”, “closed geodesics”, “Riemannian metric”

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening, Stability & uniqueness

  4. Surfaces of section for geodesic flows of closed surfaces

    Gonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli et al.Wider flows PDF

    Topics Curve shortening · Shared terms “simple closed”, “closed surfaces”, “closed geodesics”

  5. Curve shortening flows on surfaces that are not convex at infinity

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening · Shared terms “Grayson”, “closed curves”, “Gage”

  6. Curve shortening flows on rotational surfaces generated by monotone convex functions

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Long-time existence & convergence

  7. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening · Shared terms “closed curves”

  8. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows PDF

    Topics Curve shortening, Stability & uniqueness

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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. 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.