Paper · Jul 4, 2026

De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

Tim Laux, Andrea Poiatti

math.APWider flowsarXiv:2607.03930
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Abstract

We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows

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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 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 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)3Laux 2022 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)4Julin 2022 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)8Chambolle 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)1Julin 2023 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)5Bellettini 2024 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)2Poiatti 2025 Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system (Andrea Poiatti, 2026)6Poiatti 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”, “Almgren Taylor”, “Taylor Wang”

  2. Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

    Andrea PoiattiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Andrea Poiatti · Shared terms “varifold solutions”, “unconditional”

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

    Tim LauxWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Tim Laux

  4. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”, “comparison principle”, “movement scheme”

  5. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Topics Weak & singular flows · Shared terms “minimizing movements”, “Giorgi”, “Almgren Taylor”

  6. Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system

    Andrea PoiattiWider flows PDF

    Topics Weak & singular flows · Same author Andrea Poiatti · Shared terms “varifold solutions”, “global weak”, “Giorgi”

  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 · Same author Tim Laux · Shared terms “varifold solutions”, “unconditional”

  8. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

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    Topics Weak & singular flows · Shared terms “minimizing movements”, “Luckhaus Sturzenhecker”, “movement scheme”

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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.