The C∞-convergence of circle packings of bounded degree to the Riemann mapping

Abstract

Thurston conjectured that hexagonal circle packings can be used to approximate the Riemann mapping. The corresponding convergence was proven by Rodin and Sullivan. He and Schramm showed that for hexagonal circle packings the convergence is C ∞ . Here the C ∞ -convergence is generalized to the case of non-hexagonal circle packings with bounded degree. Furthermore, the estimation of the convergence rate is obtained for arbitrary order derivatives.