Paper · Oct 1, 2025

Mean curvature flow through singularities

Robert Haslhofer

math.DGWider flowsarXiv:2510.01355
PDF Abstract page

As on arXiv

Abstract

We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in . With the aim of developing a satisfying theory in higher dimensions, we then describe our recent classification of all noncollapsed singularities in . Finally, we provide a detailed discussion of open problems and conjectures.

Comments: submitted to the Proceedings of the ICM 2026

Mean curvature flowSingularities(Non)collapsingWeak & singular flows

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 Hearing the shape of ancient noncollapsed flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2021)4Du 2021 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)2Choi 2021 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)7Haslhofer 2021 A nonexistence result for rotating mean curvature flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2022)1Du 2022 Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds (Farnaz Ghanbari and Samreena, 2023)8Ghanbari 2023 Classification of ancient noncollapsed flows in ℝ⁴ (Kyeongsu Choi and Robert Haslhofer, 2024)3Choi 2024 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)5Bao 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)6Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A nonexistence result for rotating mean curvature flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics (Non)collapsing, Singularities · Same author Robert Haslhofer · Shared terms “noncollapsed”

  2. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer PDF

    Topics (Non)collapsing, Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “noncollapsed”

  3. Classification of ancient noncollapsed flows in

    Kyeongsu Choi and Robert HaslhoferWider flows PDF

    Topics (Non)collapsing, Singularities · Same author Robert Haslhofer · Shared terms “noncollapsed singularities”

  4. Hearing the shape of ancient noncollapsed flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics (Non)collapsing, Singularities · Same author Robert Haslhofer · Shared terms “noncollapsed”

  5. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer

  6. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer

  7. Uniqueness and stability of singular Ricci flows in higher dimensions

    Robert Haslhofer PDF

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “higher dimensions”

  8. Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

    Farnaz Ghanbari and SamreenaWider flows PDF

    Topics (Non)collapsing, 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.