Circle Packings in the Unit Disc

Abstract

A Bl-packing is a (branched) circle packing that ``properly covers'' the unit disc. We establish some fundamental properties of such packings. We give necessary and sufficient conditions for their existence, prove their uniqueness, and show that their underlying surfaces, known as carriers, are quasiconformally equivalent to surfaces of classical Blaschke products. We also extend our earlier approximation results of to general combinatorial patterns of tangencies in Bl-packings. Finally, a branched version of the Discrete Uniformization Theorem of Beardon and Stephenson is given.