Paper · Sep 20, 2025

Huisken's Distance Comparison Principle in Higher Codimension

Qi Sun

math.DGWider flowsarXiv:2509.16823
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Abstract

We establish a variant of Huisken's distance comparison principle for reflection symmetric immersed Curve Shortening flow in . As an application, we show that certain symmetric Curve Shortening flow with a one-to-one convex projection develops a Type I singularity and becomes asymptotically circular.

Comments: 10 pages, 3 figures

Curve shorteningSingularities

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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 A distance comparison principle for curve flows with a global forcing term (Friederike Dittberner, 2022)2Dittberner 2022 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)5Langford 2023 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)8Litzinger 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)4Bryan 2023 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)3Sun 2024 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)1Sun 2025 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)6Hebbar 2026 On asymptotically conical curve shortening flows with convex projections (Alexander Mramor, 2026)7Mramor 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 Curve Shortening Flow with Convex Projections

    Qi SunWider flows PDF

    Topics Curve shortening, Singularities · Same author Qi Sun · Shared terms “convex projection”, “immersed curve”, “asymptotically circular”

  2. A distance comparison principle for curve flows with a global forcing term

    Friederike DittbernerWider flows PDF

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

  3. Curve Shortening Flow of Space Curves with Convex Projections

    Qi SunWider flows PDF

    Topics Curve shortening · Same author Qi Sun · Shared terms “immersed curve”, “convex projection”

  4. 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 · Shared terms “distance comparison”

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

    Mat Langford and Jonathan J. ZhuWider flows PDF

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

  6. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows PDF

    Topics Curve shortening · Same author Qi Sun · Shared terms “immersed”

  7. On asymptotically conical curve shortening flows with convex projections

    Alexander MramorWider flows PDF

    Topics Curve shortening · Shared terms “convex projection”

  8. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics 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)

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.