Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps

The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variation of a theorem by Courant, Manel and Shiffman is proved and generalized. If $G$ is an $n+1$-connected bounded planar domain, $H$ is a simply connected bounded planar domain, and $P_1,P_2,\mathrm{.}\mathrm{.}\mathrm{.},P_n$ are (compact) planar convex bodies, then sets $P_j^{\mathrm{\prime }}$ can be found so that $G$ is conformally equivalent to $H-{\cup }_{j=1}^n{P}_j^{\mathrm{\prime }}$, and each $P_j^{\mathrm{\prime }}$ is either a point, or is positively homothetic to $P_j$.