Paper · Mar 6, 2026

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

Theodora Bourni, Nathan Burns, Mat Langford

math.DGWider flowsarXiv:2603.06949
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Abstract

We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in which converges in finite time to a round half-point.

Curve shorteningSingularitiesStability & uniquenessFree boundary & capillary

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 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)2Bourni 2021 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)4Bourni 2022 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)3Langford 2023 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)7Bourni 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)6Bryan 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)1Bourni 2024 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)8Nguyen 2026 On the Gage-Hamilton-Grayson Theorem for free boundary curves (Yueheng Bao and Lorenzo Sarnataro, 2026)5Bao 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 Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows PDF

    Topics Free boundary & capillary, Curve shortening · Same authors Theodora Bourni, Nathan Burns · Shared terms “convex domains”

  2. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Free boundary & capillary, Curve shortening · Same authors Theodora Bourni, Mat Langford

  3. A distance comparison principle for curve shortening flow with free boundary

    Mat Langford and Jonathan J. ZhuWider flows PDF

    Topics Free boundary & capillary, Curve shortening, Singularities · Same author Mat Langford · Shared terms “round half”, “finite time”

  4. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Free boundary & capillary · Same authors Theodora Bourni, Mat Langford

  5. On the Gage-Hamilton-Grayson Theorem for free boundary curves

    Yueheng Bao and Lorenzo SarnataroWider flows PDF

    Topics Free boundary & capillary, Curve shortening, Singularities

  6. Sharp distance comparison for curve shortening flow on the round sphere

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows PDF

    Topics Curve shortening, Singularities · Same author Mat Langford · Shared terms “finite time”

  7. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows PDF

    Topics Curve shortening, Singularities · Same author Theodora Bourni · Shared terms “finite time”

  8. High Codimension Curve Shortening Flow with Free Boundary

    Huy The Nguyen and Artemis A. VogiatziWider flows PDF

    Topics Free boundary & capillary, Curve shortening, Singularities

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.