Paper · May 26, 2025

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

Irene Gonzalvez, Alfredo Miranda, Julio D. Rossi, Jorge Ruiz-Cases

math.APWider flowsarXiv:2505.20559v3
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As on arXiv

Abstract

In this paper we introduce a new two-player zero-sum game whose value function approximates the level set formulation for the geometric evolution by mean curvature of a hypersurface. In our approach the game is played with symmetric rules for the two players and probability theory is involved (the game is not deterministic).

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Submitted
May 26, 2025
Last revised
Jul 25, 2025 (v3)
Subjects
math.AP, math.DG, math.PR
Journal
Communications in Mathematics, Volume 34 (2026), Issue 2 (Special issue: Latin American mathematics) (July 23, 2025) cm:15764
DOI
10.46298/cm.15764

Comments: arXiv admin note: text overlap with arXiv:2409.06855

Mean curvature flowWeak & singular flows

Its neighbourhood

Connected papers

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 Generic level sets in mean curvature flow are BV solutions (Anton Ullrich and Tim Laux, 2023)6Ullrich 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)8Chodosh 2023 A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation (Kuniyasu Misu, 2024)1Misu 2024 A game approach to free boundary problems of anisotropic forced mean curvature flow equations (Takuya Sato, 2025)2Sato 2025 Generic mean curvature flow with obstacles (Tim Laux and Keisuke Takasao, 2025)5Laux 2025 Integral Brakke approximate equality for an approximate mean curvature flow (Abdelmouksit Sagueni, 2025)3Sagueni 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 Online Komlós converges to mean curvature flow (Nestor Guillen and Vladimir A. Kobzar, 2026)7Guillen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation

    Kuniyasu MisuWider flows

    Topics Mean curvature flow · Shared terms “zero sum”, “function approximates”, “sum game”

  2. A game approach to free boundary problems of anisotropic forced mean curvature flow equations

    Takuya SatoWider flows

    Topics Weak & singular flows · Shared terms “set formulation”, “deterministic”, “level set”

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

    Abdelmouksit SagueniWider flows

    Topics Weak & singular flows · Shared terms “approximate”

  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

    Topics Weak & singular flows · Shared terms “approximate”

  5. Generic mean curvature flow with obstacles

    Tim Laux and Keisuke TakasaoWider flows

    Topics Weak & singular flows · Shared terms “set formulation”, “level set”

  6. Generic level sets in mean curvature flow are BV solutions

    Anton Ullrich and Tim LauxWider flows

    Topics Weak & singular flows · Shared terms “level set”

  7. Online Komlós converges to mean curvature flow

    Nestor Guillen and Vladimir A. KobzarWider flows

    Topics Mean curvature flow · Shared terms “deterministic”

  8. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows

    Topics Weak & singular flows · Shared terms “level set”

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.