The C∞-convergence of SG circle patterns to the Riemann mapping

Abstract

Thurston conjectured that the Riemann mapping function from a simply connected region onto the unit disk can be approximated by regular hexagonal packings. Schramm introduced circle patterns with combinatorics of the square grid (SG) and showed that SG circle patterns converge to meromorphic functions. He and Schramm proved that hexagonal disk packings converge in C ∞ to the Riemann mapping. In this paper we show a similar C ∞ -convergence for SG circle patterns.