Incompressibility of domain-filling circle packings

Abstract

A Jordan domain G whose boundary is decomposed into four arcs \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) is said to be a quadrilateral. A circle packing filling the quadrilateral is a finite collection \(\mathcal {P}\) of circles, which are contained in the closure of G, and touch each other as well as the four boundary arcs in a prescribed way (encoded in a so-called quad-complex). Using the concept of prime ends, we extend these definitions to arbitrary simply connected domains, and prove incompressibility of circle packings (and more general circle ensembles, which we call circle agglomerations) filling quadrilaterals. As immediate consequence we obtain uniqueness of appropriately normalized discrete conformal mappings in the spirit of Caratheodory’s theorem. Moreover, the results lead to a novel concept of the discrete conformal modulus of (planar) simplicial quad-complexes.