Paper · Feb 28, 2023

A distance comparison principle for curve shortening flow with free boundary

Mat Langford, Jonathan J. Zhu

math.DGWider flowsarXiv:2302.14258v2
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Abstract

We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in a convex planar domain. As a consequence, we are able to prove that any such flow either converges in infinite time to a (unique) "critical chord", or contracts in finite time to a "round half-point" on the boundary.

Comments: 32 pages, 2 figures; adjusted to a more precise barrier function in Theorem 5.3

Curve shorteningSingularitiesBoundaries & non-compactFree 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)3Bourni 2021 Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary (Dongyeong Ko, 2023)4Ko 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)1Bryan 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)7Bourni 2024 Huisken's Distance Comparison Principle in Higher Codimension (Qi Sun, 2025)5Sun 2025 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)2Bourni 2026 On the Gage-Hamilton-Grayson Theorem for free boundary curves (Yueheng Bao and Lorenzo Sarnataro, 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. 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 authors Mat Langford, Jonathan J. Zhu · Shared terms “distance comparison”, “chord arc”, “infinite time”

  2. 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 · Same author Mat Langford · Shared terms “round half”, “finite time”

  3. 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 author Mat Langford

  4. 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 “chord arc”, “arc profile”, “reflected chord”

  5. Huisken's Distance Comparison Principle in Higher Codimension

    Qi SunWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “distance comparison”, “comparison principle”

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

    Yueheng Bao and Lorenzo SarnataroWider flows PDF

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “embedded free”, “planar domain”, “convex planar”

  7. 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

  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 · Shared terms “reflected”, “orthogonal”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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.