Conformal uniformization of planar packings by disk packings

TL;DR

This paper proves that certain complex fractal packings, including Sierpiński and CLE carpets, can be uniformly represented by disk packings through a generalized conformal map, extending classical uniformization results.

Contribution

It introduces a new notion of packing-conformal maps and demonstrates their ability to uniformize a broad class of fractal packings, including Sierpiński and CLE carpets.

Findings

Sierpiński packings with square-summable diameters can be uniformized by disk packings.

Sierpiński carpets and some domains are also uniformizable by disk packings.

CLE carpets can be conformally uniformized by disk packings, answering a longstanding question.

Abstract

A Sierpi\'nski packing in the $2$-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpi\'nski 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 Sierpi\'nski packings, Sierpi\'nski 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.