Conformal Uniformization and Packings
نویسنده
چکیده
A new short proof is given for Brandt and Harrington's theorem about conformal uniformizations of planar nitely connected domains as domains with boundary components of speciied shapes. This method of proof generalizes to periodic domains. Letting the uniformized domains degenerate in a controlled manner, we deduce the existence of packings of speciied shapes and with speciied combinatorics. The shapes can be arbitrary smooth disks speciied up to homothety, for example. The combinatorics of the packing is described by the contacts graph, which can be spec-iied to be any nite planar graph whose vertices correspond to the shapes. This is in the spirit of Koebe's proof of the Circle Packing Theorem as a consequence of his uniformization by circle domains.
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