Paper · May 25, 2016

Anisotropic curvature flow of immersed curves

Gwenael Mercier, Matteo Novaga, Paola Pozzi

math.APWider flowsarXiv:1605.07860v2
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Abstract

We prove short-time existence of φ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.

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Submitted
May 25, 2016
Last revised
May 3, 2017 (v2)
Subjects
math.AP, math.DG

Comments: Accepted for publication in Communications in Analysis and Geometry

Other curvature flowsAnisotropic & nonlocal flowsHarmonic maps & DeTurck trick

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 and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)6Chambolle 2020 Anisotropic curvature flow of immersed networks (Heiko Kroener, Matteo Novaga and Paola Pozzi, 2020)1Kroener 2020 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)7Cesaroni 2021 Motion by crystalline-like mean curvature: a survey (Yoshikazu Giga and Norbert Požár, 2021)3Giga 2021 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)4Bellettini 2023 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)2Gößwein 2023 Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions (Giovanni Bellettini, Shokhrukh Yu. Kholmatov and Matteo Novaga, 2025)5Bellettini 2025 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)8Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Anisotropic curvature flow of immersed networks

    Heiko Kroener, Matteo Novaga and Paola PozziWider flows

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

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

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows

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

  3. Motion by crystalline-like mean curvature: a survey

    Yoshikazu Giga and Norbert PožárWider flows

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

  4. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

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

    Giovanni Bellettini, Shokhrukh Yu. Kholmatov and Matteo NovagaWider flows

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

  6. 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”

  7. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

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

  8. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Other curvature 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.

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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)