Conformal uniformization of planar packings by disk packings

A Sierpinski packing in the 2-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpinski packing by continua whose diameters are square-summable can be uniformized by a disk packing with a packing-conformal map, a notion that generalizes conformality in open sets. Being special cases of Sierpinski packings, Sierpinski carpets and some domains can be uniformized by disk packings as well. As a corollary of the main result, the conformal loop ensemble (CLE) carpets can be uniformized conformally by disk packings, answering a question of Rohde-Werness. & COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

A Sierpinski packing in the 2-sphere refers to a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We prove that any Sierpinski packing by continua whose diameters are square-summable can be uniformized by a disk packing with a packing-conformal map, which generalizes conformality in open sets. Sierpinski carpets and some domains, as special cases of Sierpinski packings, can also be uniformized by disk packings. As a corollary, the conformal loop ensemble (CLE) carpets can be conformally uniformized by disk packings, answering a question of Rohde-Werness.