Paper · May 19, 2019

Lectures on curvature flow of networks

Carlo MAntegazza, Matteo Novaga, Alessandra Pluda

math.DGWider flowsarXiv:1905.07778
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Abstract

We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.

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Submitted
May 19, 2019
Subjects
math.DG, math.AP

Comments: These are notes of a course given by Matteo Novaga within the INdAM (Istituto Nazionale di Alta Matematica) Intensive Period "Contemporary Research in elliptic PDEs and related topics", organized by Serena Dipierro at the University of Bari, from April to June 2017

Other curvature flowsSingularities

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)6Mercier 2016 Fattening and nonfattening phenomena for planar nonlocal curvature flows (Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al., 2018)3Cesaroni 2018 Anisotropic curvature flow of immersed networks (Heiko Kroener, Matteo Novaga and Paola Pozzi, 2020)2Kroener 2020 Type-0 singularities in the network flow – Evolution of trees (Carlo Mantegazza, Matteo Novaga and Alessandra Pluda, 2022)1Mantegazza 2022 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)4Gößwein 2023 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)8Bellettini 2024 A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions (Giovanni Bellettini, Virginia Lorenzini, Matteo Novaga et al., 2026)7Bellettini 2026 Lectures on Mean Curvature Flow and Related Equations (Tom Ilmanen, 2026)5Ilmanen 2026
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

    Topics Singularities · Same authors Carlo MAntegazza, Matteo Novaga, Alessandra Pluda

  2. Anisotropic curvature flow of immersed networks

    Heiko Kroener, Matteo Novaga and Paola PozziWider flows

    Topics Other curvature flows · Same author Matteo Novaga

  3. Fattening and nonfattening phenomena for planar nonlocal curvature flows

    Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al.Wider flows

    Topics Other curvature flows, Singularities · Same author Matteo Novaga · Shared terms “planar curves”

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

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows

    Same author Matteo Novaga

  5. Lectures on Mean Curvature Flow and Related Equations

    Tom IlmanenWider flows

    Topics Singularities

  6. Anisotropic curvature flow of immersed curves

    Gwenael Mercier, Matteo Novaga and Paola PozziWider flows

    Topics Other curvature flows · Same author Matteo Novaga · Shared terms “planar curves”

  7. A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions

    Giovanni Bellettini, Virginia Lorenzini, Matteo Novaga et al.Wider flows

    Topics Other curvature flows, Singularities · Same author Matteo Novaga

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

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows

    Topics 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.