Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps
Abstract
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 p_{j=1}^n P_j', and each P_j' is either a point, or is positively homothetic to P_j.
- Publication:
-
Ph.D. Thesis
- Pub Date:
- September 2007
- DOI:
- 10.48550/arXiv.0709.0710
- arXiv:
- arXiv:0709.0710
- Bibcode:
- 2007PhDT.......441S
- Keywords:
-
- Mathematics - Complex Variables;
- Mathematics - Combinatorics;
- Mathematics - Metric Geometry;
- 52C15;
- 30C20
- E-Print:
- Modified version of PhD thesis from 1990