Paper · Sep 8, 2025

Integral Brakke approximate equality for an approximate mean curvature flow

Abdelmouksit Sagueni

math.DGWider flowsarXiv:2509.06440
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Abstract

In this paper we aim to study the consistency of the mean curvature flow via discretization. We will use discretizations by volumetric varifolds, and derive a Brakke approximate equality involving the masses of the volumetric varifolds and their approximate mean curvatures.

Mean curvature flowWeak & singular flows

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow

    Blanche Buet, Gian Paolo Leonardi, Simon Masnou et al.Wider flows PDF

    Topics Weak & singular flows · Same author Abdelmouksit Sagueni · Shared terms “approximate mean”, “Brakke”, “varifold”

  2. On the avoidance principle for codimension 1 space-time Brakke flows

    Abdelmouksit SagueniWider flows PDF

    Topics Weak & singular flows · Same author Abdelmouksit Sagueni · Shared terms “Brakke”, “varifold”

  3. The Parabolic -Higgs Equations and Codimension-two Mean Curvature Flows

    Davide Parise, Alessandro Pigati and Daniel SternWider flows PDF

    Topics Weak & singular flows · Shared terms “integral Brakke”

  4. A two-player zero-sum probabilistic game that approximates the mean curvature flow

    Irene Gonzalvez, Alfredo Miranda, Julio D. Rossi et al.Wider flows PDF

    Topics Weak & singular flows · Shared terms “approximate”

  5. 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 · Shared terms “Brakke”, “varifold”

  6. Existence of BV flow via elliptic regularization

    Kiichi TashiroWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke”

  7. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows PDF

    Topics Weak & singular flows · Shared terms “integral Brakke”

  8. Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn

    J. M. Daniels-HolgateWider flows PDF

    Topics Weak & singular flows · Shared terms “integral Brakke”

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.