Paper · May 14, 2024

Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions

Eric Kim, Dohyun Kwon

math.APWider flowsarXiv:2405.08296
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Abstract

We study the motion of sets by anisotropic curvature under a volume constraint in the plane. We establish the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of Wulff shapes of equal area, the critical point of the anisotropic perimeter functional. This is an anisotropic analogue of the results in the isotropic case studied in. The novelty of our approach is in using the Cahn-Hoffman map to parametrize boundary components as small perturbations of the Wulff shape. In addition, we show that certain reflection comparison symmetries are preserved by the flat flow, which lets us obtain uniform bounds on the distance between the convergent profile and the initial data.

Comments: 30 pages, 3 figures

Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal 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 The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions (Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al., 2021)6Julin 2021 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)4Bulanyi 2025 The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus (Vedansh Arya, Daniele De Gennaro and Anna Kubin, 2025)7Arya 2025 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)2Kubin 2025 Asymptotic behaviour of the weak inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2025)5Gao 2025 Area-preserving crystalline curvature flow in two dimensions (Eric Kim, 2026)1Kim 2026 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)3Liu 2026 Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds (Ze-Yu Ye and Xiao-Liu Wang, 2026)8Ye 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 crystalline curvature flow in two dimensions

    Eric KimWider flows PDF

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Long-time existence & convergence · Same author Eric Kim · Shared terms “area preserving”, “flat flow”, “Wulff shape”

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

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

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

  3. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “Wulff shape”

  4. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “Wulff shape”

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

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “Wulff shape”

  6. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “area preserving”, “flat flow”, “disjoint”

  7. The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus

    Vedansh Arya, Daniele De Gennaro and Anna KubinWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “area preserving”, “flat flow”, “disjoint”

  8. Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

    Ze-Yu Ye and Xiao-Liu WangWider flows PDF

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows · Shared terms “area preserving”, “Wulff”

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 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.