Paper · May 24, 2017

Equality of the usual definitions of Brakke flow

Ananda Lahiri

math.DGWider flowsarXiv:1705.08789
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Abstract

In 1978 Brakke introduced the mean curvature flow in the setting of geometric measure theory. There exist multiple variants of the original definition. Here we prove that most of them are indeed equal. One central point is to correct the proof of Brakke's §3.5, where he develops an estimate for the evolution of the measure of time-dependent test functions.

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Submitted
May 24, 2017
Subjects
math.DG

Comments: 18 pages, no figures

Mean curvature flowWeak & singular flowsLimit spaces & RCD

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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 The free-boundary Brakke flow (Nick Edelen, 2016)1Edelen 2016 Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows (Eva Kopfer and Karl-Theodor Sturm, 2016)6Kopfer 2016 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)7Erbar 2025 On the avoidance principle for codimension 1 space-time Brakke flows (Abdelmouksit Sagueni, 2025)4Sagueni 2025 Integral Brakke approximate equality for an approximate mean curvature flow (Abdelmouksit Sagueni, 2025)5Sagueni 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)8Buet 2025 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)3Hervás 2026 A note on generalized BV mean curvature flow with a critical forcing term (Kiichi Tashiro, 2026)2Tashiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The free-boundary Brakke flow

    Nick EdelenWider flows

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “Brakke flows”

  2. A note on generalized BV mean curvature flow with a critical forcing term

    Kiichi TashiroWider flows

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “Brakke flows”

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

    Javier HervásWider flows

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “Brakke flows”

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

    Abdelmouksit SagueniWider flows

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

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

    Abdelmouksit SagueniWider flows

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

  6. Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

    Eva Kopfer and Karl-Theodor Sturm

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “time dependent”

  7. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “time dependent”

  8. 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

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

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 in the L² sense. 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.