Paper · May 21, 2021

Lectures on mean curvature flow of surfaces

Robert Haslhofer

math.DGarXiv:2105.10485v2
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Abstract

Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow of surfaces, with a focus on the analysis of singularities. We will see that the surfaces evolve uniquely through neck singularities and nonuniquely through conical singularities. Studying these questions, we will also learn many general concepts and methods, such as monotonicity formulas, epsilon-regularity, weak solutions, and blowup analysis that are of great importance in the analysis of a wide range of partial differential equations. These lecture notes are from summer schools at UT Austin and CRM Montreal, and also contain a detailed discussion of open problems and conjectures.

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Submitted
May 21, 2021
Last revised
Jun 15, 2024 (v2)
Subjects
math.DG, math.AP

Comments: 29 pages, includes discussion of open problems (v2: major updates in Section 5 and Section 6)

Ricci flowMean curvature flowSingularitiesWeak & singular flows

Its neighbourhood

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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 2016, 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 Diameter and curvature control under mean curvature flow (Panagiotis Gianniotis and Robert Haslhofer, 2017)8Gianniotis 2017 Ancient asymptotically cylindrical flows and applications (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al., 2019)7Choi 2019 A note on the selfsimilarity of limit flows (Beomjun Choi, Robert Haslhofer and Or Hershkovits, 2019)5Choi 2019 Heat flow on time-dependent manifolds (Beomjun Choi, Jianhui Gao, Robert Haslhofer et al., 2020)6Choi 2020 A note on blowup limits in 3d Ricci flow (Beomjun Choi and Robert Haslhofer, 2021)2Choi 2021 On κ-solutions and canonical neighborhoods in 4d Ricci flow (Robert Haslhofer, 2023)4Haslhofer 2023 Mean curvature flow through singularities (Robert Haslhofer, 2025)1Haslhofer 2025 Lectures on curve shortening flow (Robert Haslhofer, 2026)3Haslhofer 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow through singularities

    Robert HaslhoferWider flows

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “detailed discussion”, “introduction”

  2. A note on blowup limits in 3d Ricci flow

    Beomjun Choi and Robert Haslhofer

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

  3. Lectures on curve shortening flow

    Robert HaslhoferWider flows

    Topics Singularities · Same author Robert Haslhofer · Shared terms “blowup analysis”, “partial differential”, “differential equation”

  4. On -solutions and canonical neighborhoods in 4d Ricci flow

    Robert Haslhofer

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

  5. A note on the selfsimilarity of limit flows

    Beomjun Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Singularities · Same author Robert Haslhofer · Shared terms “neck singularities”, “partial differential”, “differential equation”

  6. Heat flow on time-dependent manifolds

    Beomjun Choi, Jianhui Gao, Robert Haslhofer et al.

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

  7. Ancient asymptotically cylindrical flows and applications

    Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al.Wider flows

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

  8. Diameter and curvature control under mean curvature flow

    Panagiotis Gianniotis and Robert HaslhoferWider flows

    Topics Weak & singular flows · Same author Robert Haslhofer · Shared terms “detailed”, “intrinsic”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, 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. 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.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

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