Paper · Apr 26, 2023

Some aspects of anisotropic curvature flow of planar partitions

Giovanni Bellettini, Shokhrukh Yu. Kholmatov

math.DGWider flowsarXiv:2304.13454
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Abstract

We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the polycrystalline case.

Comments: 39 pages, 14 figures

Other curvature flowsAnisotropic & nonlocal flows

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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 A flow method for a generalization of Lₚ Christofell-Minkowski problem (Boya Li, Hongjie Ju and Yannan Liu, 2022)4Li 2022 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)7Gößwein 2023 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)1Bellettini 2024 Discrete-to-continuum crystalline curvature flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2024)8Chambolle 2024 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)2Kholmatov 2024 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)5Bellettini 2024 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)6Bulanyi 2025 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)3Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “local existence”, “classical solution”, “anisotropy”

  2. Minimizing movements for forced anisotropic curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Shokhrukh Yu. Kholmatov

  3. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows

  4. A flow method for a generalization of Christofell-Minkowski problem

    Boya Li, Hongjie Ju and Yannan LiuWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows

  5. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “classical solution”, “geometric evolution”, “presence”

  6. 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 “presence”, “geometric evolution”

  7. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Topics Anisotropic & nonlocal flows

  8. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Shared terms “anisotropy”

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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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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)