Paper · Aug 5, 2025

A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

Andrea Kubin, Domenico Angelo La Manna, Enrico Pasqualetto

math.APWider flowsarXiv:2508.03134
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Abstract

In this article, we introduce a variational algorithm, in the spirit of the minimizing movements scheme, to model the volume-preserving anisotropic mean curvature flow in 2D. We show that this algorithm can be used to prove the existence of classical solutions. Moreover, we prove that this algorithm converges to the global solution of the equation.

Comments: 43 pages

Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsWeak & singular flowsLong-time existence & convergence

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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 Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem (De Gennaro Daniele, Andrea Kubin and Anna Kubin, 2022)3Daniele 2022 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)5Julin 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)4De Gennaro 2023 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)2Julin 2023 Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles (Marco Cicalese and Andrea Kubin, 2024)6Cicalese 2024 Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions (Eric Kim and Dohyun Kwon, 2024)1Kim 2024 Asymptotic behaviour of the weak inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2025)8Gao 2025 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)7Arya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions

    Eric Kim and Dohyun KwonWider flows PDF

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Long-time existence & convergence

  2. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Same author Domenico Angelo La Manna · Shared terms “volume preserving”

  3. Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

    De Gennaro Daniele, Andrea Kubin and Anna KubinWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Same author Andrea Kubin · Shared terms “volume preserving”

  4. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Andrea Kubin

  5. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “classical solution”, “movement scheme”, “minimizing movements”

  6. Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

    Marco Cicalese and Andrea KubinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Andrea Kubin · Shared terms “movement scheme”, “minimizing movements”

  7. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows, Long-time existence & convergence · Shared terms “movement scheme”, “minimizing movements”

  8. Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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.