Paper · Dec 4, 2020

Anisotropic curvature flow of immersed networks

Heiko Kroener, Matteo Novaga, Paola Pozzi

math.APWider flowsarXiv:2012.02490
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Abstract

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the norm of the anisotropic curvature blows up.

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Submitted
Dec 4, 2020
Subjects
math.AP, math-ph, math.DG

Comments: 39 pages, 1 figure

Other curvature flowsAnisotropic & nonlocal flowsStability & uniquenessPhase fields & multiphase flows

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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)2Mercier 2016 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)8Chambolle 2020 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)3Cesaroni 2021 Type-0 singularities in the network flow – Evolution of trees (Carlo Mantegazza, Matteo Novaga and Alessandra Pluda, 2022)5Mantegazza 2022 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)6Bellettini 2023 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)1Gößwein 2023 Curvature Flow of Networks with Triple Junction Drag (Yuchuan Yang and Selim Esedoglu, 2025)4Yang 2025 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)7Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Matteo Novaga, Paola Pozzi · Shared terms “triple junction”

  2. Anisotropic curvature flow of immersed curves

    Gwenael Mercier, Matteo Novaga and Paola PozziWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same authors Matteo Novaga, Paola Pozzi

  3. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Heiko Kroener, Matteo Novaga

  4. Curvature Flow of Networks with Triple Junction Drag

    Yuchuan Yang and Selim EsedogluWider flows

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows, Other curvature flows · Shared terms “triple junction”

  5. Type-0 singularities in the network flow – Evolution of trees

    Carlo Mantegazza, Matteo Novaga and Alessandra PludaWider flows

    Topics Phase fields & multiphase flows · Same author Matteo Novaga · Shared terms “triple junction”

  6. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  7. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

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

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same author Matteo Novaga

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)