Paper · Sep 9, 2026

Immortal Curve Shortening Flow on Analytic Surfaces

Aprameya Girish Hebbar, Guanhua Shao, Qi Sun

math.DGWider flowsarXiv:2609.10197
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Abstract

We show that on any closed orientable real-analytic Riemannian surface, any smoothly immersed closed immortal curve-shortening flow converges to a unique non-constant closed geodesic.

Comments: 13 pages

Curve shorteningLong-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)7Contreras 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)5Peachey 2022 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)2Sun 2024 Huisken's Distance Comparison Principle in Higher Codimension (Qi Sun, 2025)4Sun 2025 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)3Sun 2025 Non-uniqueness of geodesic limits and a question of Grayson and Gage (Shrey Aryan and Tang-Kai Lee, 2026)1Aryan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-uniqueness of geodesic limits and a question of Grayson and Gage

    Shrey Aryan and Tang-Kai LeeWider flows PDF

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

  2. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows PDF

    Topics Curve shortening · Same author Qi Sun · Shared terms “immersed”

  3. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows PDF

    Topics Curve shortening · Same author Qi Sun · Shared terms “immersed”

  4. Huisken's Distance Comparison Principle in Higher Codimension

    Qi SunWider flows PDF

    Topics Curve shortening · Same author Qi Sun · Shared terms “immersed”

  5. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

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

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Long-time existence & convergence

  7. 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 “closed orientable”, “Riemannian surface”

  8. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening

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) 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.