ON THE CONVERGENCE OF CIRCLE PACKINGS TO THE RIEMANN MAP

Rodin and Sullivan (1987) proved Thurston's conjecture that a scheme b ased on the Circle Packing Theorem converges to the Riemann mapping, t hereby providing a refreshing geometric view of Riemann's Mapping Theo rem. We now present a new proof of the Rodin-Sullivan theorem. This pr oof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin-Sullivan paper deal s with packings based on the hexagonal combinatorics. Later, quantitat ive estimates were found, which also worked for bounded valence packin gs. Here, the bounded valence assumption is unnecessary and irrelevant . 2. Our method is rather elementary, and accessible to non-experts. I n particular, quasiconformal maps are not needed. Consequently, this g ives an independent proof of Riemann's Conformal Mapping Theorem. (The Rodin-Sullivan proof uses results that rely on Riemann's Mapping Theo rem.) 3. Our approach gives the convergence of the first and second de rivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnec essary.