Paper · Aug 1, 2025

Shrinkers of the area-preserving curve-shortening flow: Existence and saddle-point property

Nikita Cernomazov

math.DGarXiv:2508.00352v2
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Abstract

We consider homothetic evolutions of the area-preserving curve-shortening flow (APCSF), that is, classical curve shortening flow with an additional non-local forcing term. By using known results on -curves, we prove the existence of non-circular shrinkers for this flow. In our first main result, we present a partial classification scheme, similar to the well-known Abresch-Langer classification for shrinkers of curve-shortening flow. Finally, we also deduce a saddle-point property for all non-circular (APCSF)-shrinkers analogous to the known saddle-point property of Abresch-Langer curves.

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Submitted
Aug 1, 2025
Last revised
Apr 12, 2026 (v2)
Subjects
math.DG
Curve shorteningVolume-preserving & constrained flowsRicci solitons

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 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)4Peachey 2022 Delayed parabolic regularity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)7Sobnack 2024 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)6Sobnack 2024 Asymptotic circularity of immortal area-preserving curvature flows (Tatsuya Miura, 2024)2Miura 2024 Star-shaped Curves under Gage's Area-preserving Flow and the CSF (Laiyuan Gao, Shicheng Zhang and Yuntao Zhang, 2024)3Gao 2024 On a non-local area-preserving curve flow (Zezhen Sun and Yuting Wu, 2025)1Sun 2025 A delayed interior area-to-height estimate for the Curve Shortening Flow (Arjun Sobnack, 2025)5Sobnack 2025 On asymptotically conical curve shortening flows with convex projections (Alexander Mramor, 2026)8Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On a non-local area-preserving curve flow

    Zezhen Sun and Yuting WuWider flows

    Topics Volume-preserving & constrained flows · Shared terms “preserving curve”, “area preserving”, “circular”

  2. Asymptotic circularity of immortal area-preserving curvature flows

    Tatsuya MiuraWider flows

    Topics Curve shortening, Volume-preserving & constrained flows · Shared terms “area preserving”, “circular”

  3. Star-shaped Curves under Gage's Area-preserving Flow and the CSF

    Laiyuan Gao, Shicheng Zhang and Yuntao ZhangWider flows

    Topics Curve shortening, Volume-preserving & constrained flows · Shared terms “area preserving”

  4. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  5. A delayed interior area-to-height estimate for the Curve Shortening Flow

    Arjun SobnackWider flows

    Topics Curve shortening

  6. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening · Shared terms “analogous”

  7. Delayed parabolic regularity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening

  8. On asymptotically conical curve shortening flows with convex projections

    Alexander MramorWider flows

    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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 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)

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.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.