Uniqueness of the limiting shape at a network singularity
Consider a finite embedded planar network $\Gamma(t)$ moving by curvature, with only triple junctions with angles of $120^\circ$ and with fixed endpoints on the boundary of a smooth strictly convex domain. At an interior singular point $(x_0, T)$ set $$\widetilde\Gamma(t) = \frac{\Gamma(t) - x_0}{\sqrt{2(T - t)}}.$$ Along suitable sequences $t_j \uparrow T$ the networks $\widetilde\Gamma(t_j)$ converge locally to a self-similarly shrinking network, possibly degenerate and possibly with multiplicity, and a priori the limit depends on the sequence. When one such limit is a multiplicity-one line, White's local regularity theorem shows that the flow is smooth near $x_0$ up to time $T$ (so $x_0$ is not a singular point), and the limit is unique (Mantegazza, Novaga, Pluda and Schulze). Pluda and Pozzetta proved uniqueness of nondegenerate compact blow-up limits, using a Łojasiewicz–Simon gradient inequality for the length of networks.
Problem. Is the limiting shrinker independent of the sequence $t_j \uparrow T$ used to obtain it? Precisely, must all locally convergent subsequences of the rescaled networks have the same limiting arclength varifold, including its integer multiplicities? No rotation of the limit is allowed when comparing subsequences.
Source: Mantegazza, Novaga, Pluda and Schulze (arXiv 2016; Astérisque 2024), the uniqueness of blow-up assumption U, Open Problem 8.25 of the Astérisque version (7.25 in the April 2024 author text); stated for limiting arclength varifolds by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, blow-up limits, curvature flows, planar networks
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},
}
@misc{pluda2022,
author = {Pluda, Alessandra and Pozzetta, Marco},
title = {On the uniqueness of nondegenerate blowups for the motion by curvature of networks},
year = {2022},
eprint = {2204.11675},
archivePrefix = {arXiv},
}
@article{pluda2024,
author = {Pluda, Alessandra and Pozzetta, Marco},
title = {{{\L}}ojasiewicz--{S}imon inequalities for minimal networks: stability and convergence},
journal = {Math. Ann.},
volume = {389},
number = {3},
year = {2024},
pages = {2729--2782},
doi = {10.1007/s00208-023-02714-7},
eprint = {2204.08921},
archivePrefix = {arXiv},
}
@misc{aim-network-tangent-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the limiting shape at a network singularity}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/159-network-tangent-uniqueness.md},
note = {Entry 159 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-00881, "Uniqueness of the limiting shape at a network singularity", clopen.solutions.