Paper · Jan 29, 2026

Lectures on Mean Curvature Flow and Related Equations

Tom Ilmanen

math.DGWider flowsarXiv:2601.21952
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Abstract

The aim in these lectures is to study singularity formation, nonuniqueness, and topological change in motion by mean curvature.

Comments: 63 pages. These unpublished lecture notes were written by Tom Ilmanen (1961-2025) in 1995. They are posted on arXiv with the permission of his family.

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 Distributional solutions to mean curvature flow (Tim Laux, 2021)8Laux 2021 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)7Lee 2023 Singularity formation along the line bundle mean curvature flow (Yu Hin Chan and Adam Jacob, 2023)5Chan 2023 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)2Ilmanen 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)4Li 2024 Singularities of Mean Curvature Flow of Surfaces (Tom Ilmanen, 2026)1Ilmanen 2026 Lectures on curve shortening flow (Robert Haslhofer, 2026)3Haslhofer 2026 Increase in topological complexity along the mean curvature flow (Natasha Diederen, 2026)6Diederen 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 Mean Curvature Flow of Surfaces

    Tom Ilmanen PDF

    Topics Singularities · Same author Tom Ilmanen

  2. Fattening in mean curvature flow

    Tom Ilmanen and Brian White PDF

    Topics Singularities · Same author Tom Ilmanen

  3. Lectures on curve shortening flow

    Robert HaslhoferWider flows PDF

    Topics Singularities

  4. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Shared terms “singularity formation”

  5. Singularity formation along the line bundle mean curvature flow

    Yu Hin Chan and Adam JacobWider flows PDF

    Topics Singularities · Shared terms “singularity formation”

  6. Increase in topological complexity along the mean curvature flow

    Natasha DiederenWider flows PDF

    Topics Singularities

  7. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Singularities

  8. Distributional solutions to mean curvature flow

    Tim LauxWider flows PDF

    Topics Mean curvature flow

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.