On the convergence of circle packings to the Riemann map

Abstract.

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues.

1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant.

2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.)

3. Our approach gives the convergence of the first and second derivatives, 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 unnecessary.