Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps

Details Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps

2007-09-05

arxiv.org/abs/0709.0710v1

Ph. D. thesis. Princeton University (1990)

Mathematics

Complex Variables

Modified version of PhD thesis from 1990

Scientific paper

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 $P_1,P_2,...,P_n$ are (compact) planar convex bodies, then sets $P_j'$ can be found so that $G$ is conformally equivalent to $H-\cup_{j=1}^n P_j'$, and each $P_j'$ is either a point, or is positively homothetic to $P_j$.

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