Paper · Feb 24, 2026

High Codimension Curve Shortening Flow with Free Boundary

Huy The Nguyen, Artemis A. Vogiatzi

math.DGWider flowsarXiv:2602.20865
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Abstract

We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound . Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.

Curve shorteningSelf-shrinkers & translatorsSingularitiesFree boundary & capillary

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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)6Bourni 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)7Bourni 2022 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)2Langford 2023 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)4Vogiatzi 2023 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)1Litzinger 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)5Bourni 2024 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)3Bourni 2026 On the Gage-Hamilton-Grayson Theorem for free boundary curves (Yueheng Bao and Lorenzo Sarnataro, 2026)8Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “codimension curve”, “high codimension”, “low entropy”

  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 “reflected”, “orthogonal”

  3. 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

  4. Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

    Artemis A. Vogiatzi and Huy T. NguyenWider flows PDF

    Topics Singularities · Same authors Huy The Nguyen, Artemis A. Vogiatzi · Shared terms “high codimension”

  5. 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

  6. 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

  7. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows PDF

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

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

    Yueheng Bao and Lorenzo SarnataroWider 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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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.