Paper · Oct 15, 2026

Generic Mean Curvature Flow With Rough Weights

Fabius Krämer, Tim Laux, Keisuke Takasao

math.APWider flowsarXiv:2610.21545
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Abstract

We prove the generic existence of mean curvature flows in spaces with rough weights. Our weak solutions are both unit-density Brakke flows and distributional solutions. Such a generic statement is optimal in the sense that we consider weights that are too rough to have well-defined traces on every hypersurface.

Comments: 27 pages, 1 figure

Mean curvature flowWeak & singular flows

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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)5Laux 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)8Takasao 2022 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)6Laux 2022 Generic level sets in mean curvature flow are BV solutions (Anton Ullrich and Tim Laux, 2023)2Ullrich 2023 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)7Chiesa 2025 Generic mean curvature flow with obstacles (Tim Laux and Keisuke Takasao, 2025)1Laux 2025 Equivalence of weak solution concepts for mean curvature flow (Tim Laux and Anton Ullrich, 2025)3Laux 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)4Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generic mean curvature flow with obstacles

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Same authors Tim Laux, Keisuke Takasao · Shared terms “generic mean”, “distributional solutions”

  2. 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”, “Brakke flows”

  3. Equivalence of weak solution concepts for mean curvature flow

    Tim Laux and Anton UllrichWider flows PDF

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

  4. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Same authors Tim Laux, Keisuke Takasao · Shared terms “Brakke”

  5. Distributional solutions to mean curvature flow

    Tim LauxWider flows PDF

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

  6. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows PDF

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

  7. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke flows”

  8. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Same author Keisuke Takasao · Shared terms “distributional”

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)

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.