Riemann mapping theorems for Beltrami equations by circle packings

Abstract

We use circle packing techniques to construct approximate solutions to the generalized Beltrami equations with simply and multiply connected regions in the plane. We show convergence of the approximate solutions. This gives a constructive proof for the existence of quasiconformal mappings with a given pair of complex dilations.

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

“…This conjecture was proved in [10]. Refinements and generalizations were given in [4][5][6][7][8]11,13], etc.…”

Approximation of quasiconformal mappings by SG circle patterns

“…For the study of the discrete approximation of quasiconformal mappings, He [16] used circle packings to construct the approximating solutions to Beltrami equations; Lan and Dai [17] applied similar techniques to discuss the discrete approximation of quasiconformal mappings with two complex dilations. But these methods of constructing approximating solutions are not direct, i.e.…”