Paper · Oct 7, 2026

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

Yueheng Bao, Lorenzo Sarnataro

math.DGWider flowsarXiv:2610.09557
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Abstract

We provide an alternative and independent proof of the analogue of the Gage-Hamilton-Grayson Theorem for embedded free boundary curve shortening flows in a convex planar domain established by Langford-Zhu. Our proof is based on blowup arguments introduced by B. White in his work on mean-convex mean curvature flow.

Mean curvature flowCurve shorteningSingularitiesFree 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)4Bourni 2021 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)2Langford 2023 Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary (Dongyeong Ko, 2023)5Ko 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)3Bourni 2024 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)8Nguyen 2026 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)1Bourni 2026 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)7Bao 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)6Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows PDF

    Topics Free boundary & capillary, Curve shortening, Singularities

  2. 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 · Shared terms “embedded free”, “planar domain”, “convex planar”

  3. 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

  4. 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

  5. Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary

    Dongyeong KoWider flows PDF

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “Grayson Theorem”, “Langford Zhu”

  6. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Free boundary & capillary, Singularities · Same author Yueheng Bao · Shared terms “mean convex”

  7. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Free boundary & capillary, Singularities · Same author Yueheng Bao · Shared terms “mean convex”, “White”

  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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 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.