Paper · Oct 9, 2023

Stability analysis for the anisotropic curve shortening flow of planar networks

Michael Gößwein, Matteo Novaga, Paola Pozzi

math.APWider flowsarXiv:2310.05596
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Abstract

In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are -close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.

Curve shorteningAnisotropic & nonlocal flowsStability & uniquenessPhase fields & multiphase flows

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 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 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)6Cesaroni 2021 A novel finite element approximation of anisotropic curve shortening flow (Klaus Deckelnick and Robert Nürnberg, 2021)5Deckelnick 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)7Cesaroni 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)8Peachey 2022 Type-0 singularities in the network flow – Evolution of trees (Carlo Mantegazza, Matteo Novaga and Alessandra Pluda, 2022)1Mantegazza 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)2Bourni 2023 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)3Bellettini 2024 Curvature Flow of Networks with Triple Junction Drag (Yuchuan Yang and Selim Esedoglu, 2025)4Yang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type-0 singularities in the network flow – Evolution of trees

    Carlo Mantegazza, Matteo Novaga and Alessandra PludaWider flows PDF

    Topics Phase fields & multiphase flows, Curve shortening · Same author Matteo Novaga · Shared terms “junctions”

  2. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows PDF

    Topics Curve shortening · Shared terms “anisotropic curve”

  3. Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows · Shared terms “planar network”

  4. Curvature Flow of Networks with Triple Junction Drag

    Yuchuan Yang and Selim EsedogluWider flows PDF

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

  5. A novel finite element approximation of anisotropic curve shortening flow

    Klaus Deckelnick and Robert NürnbergWider flows PDF

    Topics Anisotropic & nonlocal flows, Curve shortening, Stability & uniqueness

  6. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows PDF

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

  7. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows PDF

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same author Matteo Novaga · Shared terms “initial data”

  8. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening, Stability & uniqueness

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.

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 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) 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)

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.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.