Domain-filling circle packings

Abstract

According to a classical theorem of Constantin Carathéodory, any Jordan domain G admits a unique conformal mapping onto the unit disk D such that three distinguished boundary points of G have prescribed images on ∂D. This result can be extended to general domains when the role of boundary points is taken over by prime ends. In this paper we prove a discrete counterpart of Carathéodory’s theorem in the framework of circle packing. A trilateral is a domain whose (intrinsic) boundary is decomposed into three arcs α, β, and γ (of prime ends). Using Sperner’s lemma, we prove existence of circle packings and more general circle agglomerations filling arbitrary (bounded) simply connected trilaterals. In order to get complete results we have to admit (in some exceptional cases) that the packings contain degenerate circles. We study in detail which properties of the domain G and the underlying complex of the packing guarantee uniqueness and exclude degeneracy.