Paper · Sep 25, 2026

General regularity obstructions for the arrival time equation

Yiqi Huang, Jingze Zhu

math.DGWider flowsarXiv:2609.31599
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Abstract

We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be . Our companion paper proves that every such arrival time is for any . Hence the obstruction at order gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.

Mean curvature flowCurve shorteningSingularities

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 Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)5Minarčík 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)3Litzinger 2023 Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow (Gabriel Khan, 2023)7Khan 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)4Bryan 2023 Arnold-Thom conjecture for the arrival time of surfaces (Tang-Kai Lee and Jingze Zhu, 2024)2Lee 2024 Delayed parabolic regularity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 Huisken's Distance Comparison Principle in Higher Codimension (Qi Sun, 2025)6Sun 2025 Optimal regularity for the arrival time equation (Yiqi Huang and Jingze Zhu, 2026)1Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Optimal regularity for the arrival time equation

    Yiqi Huang and Jingze ZhuWider flows PDF

    Topics Mean curvature flow · Same authors Yiqi Huang, Jingze Zhu · Shared terms “arrival time”, “time equation”, “time function”

  2. Arnold-Thom conjecture for the arrival time of surfaces

    Tang-Kai Lee and Jingze ZhuWider flows PDF

    Topics Singularities · Same author Jingze Zhu · Shared terms “arrival time”, “time function”, “degenerate”

  3. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics Curve shortening, Singularities

  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

  5. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows PDF

    Topics Curve shortening, Singularities

  6. Huisken's Distance Comparison Principle in Higher Codimension

    Qi SunWider flows PDF

    Topics Curve shortening, Singularities

  7. Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

    Gabriel KhanWider flows PDF

    Topics Curve shortening, Singularities

  8. Delayed parabolic regularity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening

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)

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.