The convergence of inversive distance circle packings to the Riemann mapping

TL;DR

This paper proves that inversive distance circle packings, a generalization of Thurston's circle packings, converge to the Riemann mapping for Jordan domains, confirming a conjecture by Bowers and Stephenson.

Contribution

It establishes the solvability of curvature prescription problems for inversive distance circle packings, confirming their convergence to the Riemann mapping.

Findings

Proof of Bowers-Stephenson's conjecture for Jordan domains

Solvability theorem for combinatorial curvature problems

Convergence of discrete conformal maps to the Riemann mapping

Abstract

Bowers and Stephenson introduced the notion of inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured that discrete conformal maps induced by inversive distance circle packings converge to the Riemann mapping. In this paper, we prove Bowers-Stephenson's conjecture for Jordan domains by establishing a solvability theorem of certain prescribing combinatorial curvature problems for inversive distance circle packings.