Quasiconformal uniformization and extension for planar trees
Abstract
A classical uniformization theorem of Ahlfors from 1963 asserts that every bounded turning planar arc is the image of the unit segment under a quasiconformal self-map of the plane. We prove an analogous uniformization theorem for planar trees by showing that every planar quasiconformal tree with uniform branch separation is quasiconformally equivalent to a quasiconvex planar tree with uniform branch separation and Hausdorff dimension arbitrarily close to 1. Along the way, we show that every quasisymmetric (resp. bi-Lipschitz) homeomorphism between planar quasiconformal trees that respects the counterclockwise orientation of the branches at every branch point can be extended quasisymmetrically (resp. bi-Lipschitz) to the whole plane. As a corollary of the extension theorem we construct a large class of quadrivalent quasiconformal trees, which includes the Vicsek fractal, so that each tree in this class is the Julia set of a uniformly quasiregular map of ℝ2.
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BibTeX
@misc{ioannidis2026quasiconformal,
title = {{Quasiconformal uniformization and extension for planar trees}},
author = {Fotis Ioannidis and Vyron Vellis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14236},
eprint = {2610.14236},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14236},
note = {CC0},
}
RIS
TY - UNPB
TI - Quasiconformal uniformization and extension for planar trees
AU - Fotis Ioannidis
AU - Vyron Vellis
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.14236
UR - https://arxiv.org/abs/2610.14236
AB - A classical uniformization theorem of Ahlfors from 1963 asserts that every bounded turning planar arc is the image of the unit segment under a quasiconformal self-map of the plane. We prove an analogous uniformization theorem for planar trees by showing that every planar quasiconformal tree with uniform branch separation is quasiconformally equivalent to a quasiconvex planar tree with uniform branch separation and Hausdorff dimension arbitrarily close to 1. Along the way, we show that every quasisymmetric (resp. bi-Lipschitz) homeomorphism between planar quasiconformal trees that respects the counterclockwise orientation of the branches at every branch point can be extended quasisymmetrically (resp. bi-Lipschitz) to the whole plane. As a corollary of the extension theorem we construct a large class of quadrivalent quasiconformal trees, which includes the Vicsek fractal, so that each tree in this class is the Julia set of a uniformly quasiregular map of ℝ^2.
ER -