CONFORMAL UNIFORMIZATION AND PACKINGS

A new short proof is given for Brandt and Harrington's theorem about c onformal uniformizations of planar finitely connected domains as domai ns with boundary components of specified shapes. This method of proof generalizes to periodic domains. Letting the uniformized domains degen erate in a controlled manner, we deduce the existence of packings of s pecified shapes and with specified combinatorics. The shapes can be ar bitrary smooth disks specified up to homothety, for example. The combi natorics of the packing is described by the contacts graph, which can be specified to be any finite planar graph whose vertices correspond t o the shapes. This is in the spirit of Koebe's proof of the Circle Pac king Theorem as a consequence of his uniformization by circle domains.