As on arXiv
Abstract
The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a Type-0 singularity, in contrast to the well known Type-I and Type-II ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree-like networks till the first singular time, under the assumption that all the tangents flows have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results by Ilmanen-Neves-Schulze/Lira-Mazzeo-Pluda-Saez to restart the flow after the singularity.
Comments: arXiv admin note: text overlap with arXiv:1611.08254
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.
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Stability analysis for the anisotropic curve shortening flow of planar networks
Topics Phase fields & multiphase flows, Curve shortening · Same author Matteo Novaga · Shared terms “junctions”
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Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions
Topics Phase fields & multiphase flows · Shared terms “junctions”
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Curvature Flow of Networks with Triple Junction Drag
Topics Phase fields & multiphase flows · Shared terms “junctions”
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Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Topics Curve shortening, Singularities · Shared terms “contrast”
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Singularities of low entropy high codimension curve shortening flow
Topics Curve shortening, Singularities
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Long-term behavior of curve shortening flow in
Topics Curve shortening, Singularities · Shared terms “contrast”
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Long time behavior for a curvature flow of networks related to grain bundary motion with the effect of lattice misorientations
Topics Phase fields & multiphase flows · Shared terms “junctions”
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Singularities of Curve Shortening Flow with Convex Projections
Topics Curve shortening, Singularities · Shared terms “tangent flow”
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.
In the guidebook
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.
- 12C.4Loops, Disks and Curve Shrinking
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.
- 6A.13Self-Similar Variables and Monotonicity
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.