Paper · Sep 8, 2025

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

Abdelmouksit Sagueni

math.DGWider flowsarXiv:2509.06441
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Abstract

We prove that the spacetime Brakke flow constructed by Buet et al. is non-trivial as long as the initial varifold is a union of boundaries of domains of finite perimeter. In the codimension 1 setting, we show that, starting from a smooth boundary, the support of the mass measure of the spacetime Brakke flow coincides with the support of the classical mean curvature flow.

Mean curvature flowWeak & singular flows

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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 Existence of BV flow via elliptic regularization (Kiichi Tashiro, 2023)6Tashiro 2023 The Parabolic U(1)-Higgs Equations and Codimension-two Mean Curvature Flows (Davide Parise, Alessandro Pigati and Daniel Stern, 2023)5Parise 2023 An Intersection Principle for Mean Curvature Flow (Tang-Kai Lee and Alec Payne, 2025)3Lee 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)7Chiesa 2025 Integral Brakke approximate equality for an approximate mean curvature flow (Abdelmouksit Sagueni, 2025)2Sagueni 2025 Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow (Blanche Buet, Gian Paolo Leonardi, Simon Masnou et al., 2025)1Buet 2025 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)8Hervás 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)4Bao 2026
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 “spacetime Brakke”, “Brakke flows”, “varifold”

  2. Integral Brakke approximate equality for an approximate mean curvature flow

    Abdelmouksit SagueniWider flows PDF

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

  3. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows PDF

    Topics Weak & singular flows · Shared terms “avoidance principle”, “Brakke flows”

  4. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Weak & singular flows · Shared terms “avoidance”, “Brakke flows”

  5. 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 “Brakke flows”

  6. Existence of BV flow via elliptic regularization

    Kiichi TashiroWider flows PDF

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

  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 · Shared terms “Brakke flows”, “varifold”

  8. Spacelike Brakke flows with boundary in pseudo-Euclidean space

    Javier HervásWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”, “varifold”

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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.