Paper · Jun 18, 2025

Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions

Giovanni Bellettini, Shokhrukh Yu. Kholmatov, Matteo Novaga

math.APWider flowsarXiv:2506.15869
View PDF View HTML arXiv page

As on arXiv

Abstract

Given a planar crystalline anisotropy, we study the crystalline elastic flow of immersed polygonal curves, possibly also unbounded. Assuming that the segments evolve by parallel translation (as it happens in the standard crystalline curvature flow), we prove that a unique regular flow exists until a maximal time when some segments having zero crystalline curvature disappear. Furthermore, for closed polygonal curves, we analyze the behaviour at the maximal time, and show that it is possible to restart the flow finitely many times, yielding a globally in time evolution, that preserves the index of the curve. Next, we investigate the long-time properties of the flow using a Lojasiewicz-Simon-type inequality, and show that, as time tends to infinity, the flow fully converges to a stationary curve. We also provide a complete classification of the stationary solutions and a partial classification of the translating solutions in the case of the square anisotropy.

View PDF View HTML arXiv page
Submitted
Jun 18, 2025
Subjects
math.AP, math.CA

Comments: 39 pages, 18 figures

Other curvature flowsAnisotropic & nonlocal flowsSelf-shrinkers & translatorsLong-time existence & convergence

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 2016, 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 Anisotropic curvature flow of immersed curves (Gwenael Mercier, Matteo Novaga and Paola Pozzi, 2016)5Mercier 2016 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)8Chambolle 2017 Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities (Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. Kholmatov, 2020)3Bellettini 2020 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)7Chambolle 2020 Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions (Annalisa Cesaroni, Heiko Kröner and Matteo Novaga, 2021)6Cesaroni 2021 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)4Cesaroni 2021 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)2Bellettini 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
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

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “crystalline”, “long time”, “anisotropy”

  2. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

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

  3. Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

    Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “anisotropy”

  4. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Self-shrinkers & translators · Same author Matteo Novaga · Shared terms “crystalline”, “anisotropy”

  5. Anisotropic curvature flow of immersed curves

    Gwenael Mercier, Matteo Novaga and Paola PozziWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Matteo Novaga · Shared terms “crystalline curvature”

  6. Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

    Annalisa Cesaroni, Heiko Kröner and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Same author Matteo Novaga · Shared terms “anisotropy”

  7. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “crystalline”

  8. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “crystalline”, “anisotropy”, “possibly”

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.

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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)