Conformal uniformization of planar packings by disk packings

Abstract

A Sierpiński packing in the $2$-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpiński 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 Sierpiński packings, Sierpiński carpets and some domains and 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.

arXiv comment48 pages, 7 figures

References in corpus (7)

• Uniformization of Sierpiński carpets in the plane
• Polyhedral approximation of metric surfaces and applications to uniformization
• Rigidity theorems for circle domains
• Rigidity and continuous extension for conformal maps of circle domains
• Metric definition of quasiconformality and exceptional sets
• Monotone Sobolev functions in planar domains: level sets and smooth approximation
• On the Hausdorff dimension of the residual set of a packing by smooth curves

Cited by in corpus (1)

• Rigidity and continuous extension for conformal maps of circle domains