Paper · Aug 6, 2026

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

Kiichi Tashiro

math.APWider flowsarXiv:2608.05760
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Abstract

In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

Comments: 16 pages, 1 figure, comment welcome!

Mean curvature flowWeak & singular flowsLimit spaces & RCD

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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 Existence of BV flow via elliptic regularization (Kiichi Tashiro, 2023)1Tashiro 2023 On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds (Maxwell Stolarski, 2023)8Stolarski 2023 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)2Tashiro 2024 Existence of curvature flow with forcing in a critical Sobolev space (Yuning Liu and Yoshihiro Tonegawa, 2024)5Liu 2024 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)6Sagueni 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)4Buet 2025 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)3Hervás 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of BV flow via elliptic regularization

    Kiichi TashiroWider flows PDF

    Topics Weak & singular flows · Same author Kiichi Tashiro · Shared terms “Brakke flows”, “Tonegawa”, “Stuvard”

  2. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows PDF

    Topics Weak & singular flows · Same author Kiichi Tashiro · Shared terms “Brakke”

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

    Javier HervásWider flows PDF

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

  4. 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 · Shared terms “Brakke flows”, “Tonegawa”, “flow satisfies”

  5. Existence of curvature flow with forcing in a critical Sobolev space

    Yuning Liu and Yoshihiro TonegawaWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke”

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

    Abdelmouksit SagueniWider flows PDF

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

  7. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz PDF

    Topics Limit spaces & RCD, Weak & singular flows

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

    Maxwell StolarskiWider flows PDF

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

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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) 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.