Paper · Aug 18, 2021

Distributional solutions to mean curvature flow

Tim Laux

math.APWider flowsarXiv:2108.08347
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Abstract

These lecture notes aim to present some of the ideas behind the recent (conditional) existence and (weak-strong) uniqueness theory for mean curvature flow. Focusing on the simplest case of the evolution of a single closed hypersurface allows for a self-contained and concise presentation, which is accessible for beginning graduate students with some background in PDEs and only requires basic measure theory.

Comments: 38 pages, 3 figures

Mean curvature flowWeak & singular flowsStability & uniqueness

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 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)7Hensel 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)8Hensel 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)1Laux 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)6Laux 2022 Generic level sets in mean curvature flow are BV solutions (Anton Ullrich and Tim Laux, 2023)4Ullrich 2023 Generic mean curvature flow with obstacles (Tim Laux and Keisuke Takasao, 2025)5Laux 2025 Equivalence of weak solution concepts for mean curvature flow (Tim Laux and Anton Ullrich, 2025)2Laux 2025 Generic Mean Curvature Flow With Rough Weights (Fabius Krämer, Tim Laux and Keisuke Takasao, 2026)3Krämer 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “distributional solutions”

  2. Equivalence of weak solution concepts for mean curvature flow

    Tim Laux and Anton UllrichWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “distributional solutions”

  3. Generic Mean Curvature Flow With Rough Weights

    Fabius Krämer, Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Same author Tim Laux · Shared terms “distributional solutions”

  4. Generic level sets in mean curvature flow are BV solutions

    Anton Ullrich and Tim LauxWider flows PDF

    Topics Weak & singular flows · Same author Tim Laux · Shared terms “distributional solutions”

  5. Generic mean curvature flow with obstacles

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Same author Tim Laux · Shared terms “distributional solutions”

  6. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “distributional solutions”

  7. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”

  8. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

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

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

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.