Multiplicity one at planar network singularities
Let $\Gamma(t)$, $0 \le t < T$, be a smooth embedded finite network in a bounded smooth strictly convex planar domain, whose interior vertices are triple junctions with angles of $120^\circ$, whose endpoints on the boundary are fixed, and each of whose edges moves with normal velocity equal to its curvature. Let $T < \infty$ be its first singular time and $x_0$ an interior singular point. For $\lambda_j \to \infty$ consider the parabolic rescalings $$\Gamma_j(s) = \lambda_j \bigl(\Gamma(T + s/\lambda_j^2) - x_0\bigr), \qquad s < 0.$$ Multiplicity counts how many rescaled arcs converge to the same limiting arc; it is retained by taking limits of the associated arclength varifolds. Mantegazza, Novaga and Pluda proved that every such limit is embedded with multiplicity one when the initial network has at most two triple junctions. Mantegazza, Novaga, Pluda and Schulze proved the same when the triple junctions stay uniformly far from each other and from the endpoints during the flow (in which case no singularity forms at all), and much of their description of the flow at and after a singular time assumes multiplicity one.
Problem. Must every self-similarly shrinking limit obtained from the rescalings $\Gamma_j$ be embedded and have multiplicity one?
Source: Mantegazza, Novaga, Pluda and Schulze (arXiv 2016; Astérisque 2024), the multiplicity-one conjecture M1, Open Problem 10.1 of the Astérisque version (9.1 in the April 2024 author text), for limits of parabolic rescalings or of Huisken's rescaling; AIM — Open Applied Problems states it for parabolic rescalings.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, curvature flows, planar networks, self-similar shrinkers
References
@book{mantegazza2024,
author = {Mantegazza, Carlo and Novaga, Matteo and Pluda, Alessandra and Schulze, Felix},
title = {Evolution of networks with multiple junctions},
series = {Ast{\'e}risque},
volume = {452},
publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
address = {Paris},
year = {2024},
isbn = {978-2-85629-997-5},
eprint = {1611.08254},
archivePrefix = {arXiv},
}
@article{mantegazza2017,
author = {Mantegazza, Carlo and Novaga, Matteo and Pluda, Alessandra},
title = {Motion by curvature of networks with two triple junctions},
journal = {Geom. Flows},
volume = {2},
number = {1},
year = {2017},
pages = {18--48},
doi = {10.1515/geofl-2016-0002},
eprint = {1606.08011},
archivePrefix = {arXiv},
}
@misc{aim-network-multiplicity-one,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Multiplicity one at planar network singularities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/158-network-multiplicity-one.md},
note = {Entry 158 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00879, "Multiplicity one at planar network singularities", clopen.solutions.