The C1 Convergence of Hexagonal Disk Packings to the Riemann Map
نویسنده
چکیده
TO THE RIEMANN MAP Zheng-Xu He and Oded Schramm Abstract. Let $ C be a simply connected domain. The Rodin-SullivanTheorem states that a sequence of disk packings in the unit disk U converges, in a well de ned sense, to a conformal map from to U . Moreover, it is known that the rst and second derivatives converge as well. Here, it is proven that for hexagonal disk packings the convergence is C1 . This is done by studying M obius invariants of the disk packings that are discrete analogs of the Schwarzian derivative. As a consequence, the rst n derivatives of the conformal map can be approximated by quantities which depend on the positions of the centers of some n + 1 consecutive disks in the packing.
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