Paper · Oct 13, 2025

Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

Jiuzhou Huang

math.DGWider flowsarXiv:2510.11430
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we construct a family of mean curvature flow which converges to an area minimizing, strictly stable hypercone after type I rescaling, and converges to the Hardt-Simon foliation of the cone after a type II rescaling provided the cone satisfies some technique conditions. The difference from Velázquez's previous results is that we drop the symmetry condition on the cone.

View PDF View HTML arXiv page
Submitted
Oct 13, 2025
Subjects
math.DG
Mean curvature flowSingularities

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 Type II smoothing in mean curvature flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2021)6Angenent 2021 Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)4Sun 2022 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)8Chu 2023 On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds (Maxwell Stolarski, 2023)1Stolarski 2023 Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature (Zichang Liu, 2024)5Liu 2024 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)3Choi 2024 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)7Chodosh 2024 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)2Park 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows

    Topics Singularities · Shared terms “Hardt Simon”, “area minimizing”

  2. Ancient mean curvature flow asymptotic to a minimal quadratic cone

    Junyoung ParkWider flows

    Topics Mean curvature flow · Shared terms “Simon foliation”, “Hardt Simon”

  3. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Mean curvature flow · Same author Jiuzhou Huang

  4. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

    Ao Sun and Jinxin XueWider flows

    Topics Singularities · Shared terms “flow converging”

  5. Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

    Zichang LiuWider flows

    Topics Singularities · Shared terms “Velazquez”

  6. Type II smoothing in mean curvature flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities · Shared terms “Velazquez”

  7. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows

    Topics Singularities

  8. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows

    Topics 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.

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)

In the guidebook

  1. 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.

  2. 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.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.