Paper · Oct 20, 2025

The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

Yiqi Huang, Wenshuai Jiang

math.DGWider flowsarXiv:2510.17060v2
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Abstract

We show that the intrinsic diameter of mean curvature flow in is uniformly bounded as one approaches the first singular time . This confirms the bounded diameter conjecture of Haslhofer. In addition, we establish several sharp quantitative estimates: the second fundamental form has uniformly bounded -norm on each time slice, belongs to the weak space on the space-time region, and the singular set has finite -Hausdorff measure. All of the results are optimal due to the marriage ring example and our results do not require any convexity assumptions on the surfaces. Furthermore, our arguments extend naturally to flows through singularities, yielding the same sharp estimates.

Mean curvature flowSingularitiesWeak & singular flows

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows PDF

    Topics Weak & singular flows, Singularities · Shared terms “Haslhofer”

  2. On the rate of convergence of cylindrical singularity in mean curvature flow

    Yiqi Huang and Xinrui ZhaoWider flows PDF

    Topics Singularities · Same author Yiqi Huang · Shared terms “singular set”

  3. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows PDF

    Topics Weak & singular flows, Singularities · Shared terms “Hausdorff”

  4. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows PDF

    Topics Weak & singular flows, Singularities

  5. Existence of curvature flow with forcing in a critical Sobolev space

    Yuning Liu and Yoshihiro TonegawaWider flows PDF

    Topics Weak & singular flows, Singularities · Shared terms “Hausdorff measure”

  6. General regularity obstructions for the arrival time equation

    Yiqi Huang and Jingze ZhuWider flows PDF

    Topics Singularities · Same author Yiqi Huang

  7. Mean curvature flow through singularities

    Robert HaslhoferWider flows PDF

    Topics Weak & singular flows, Singularities

  8. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue PDF

    Topics Weak & singular flows, Singularities

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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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.