On rigidity and convergence of circle patterns

TL;DR

This paper establishes rigidity results for circle patterns with the same combinatorics and intersection angles, showing they are related by hyperbolic isometries and that discrete conformal maps approximate Riemann maps.

Contribution

It proves new rigidity theorems for circle patterns in the plane and disk, and demonstrates convergence of discrete conformal maps to classical Riemann maps.

Findings

Circle patterns with same combinatorics and intersection angles are related by hyperbolic isometries.

Discrete conformal maps from circle patterns converge to Riemann maps under certain conditions.

Rigidity holds for circle patterns in both the disk and the complex plane with bounded exterior angles.

Abstract

Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a hyperbolic isometry. Furthermore, we prove an analogous rigidity statement for the complex plane if all exterior intersection angles of neighboring circles are uniformly bounded away from $0$. Finally, we study a sequence of two circle patterns with the same combinatorics each of which approximates a given simply connected domain. Assume that all kites are convex and all angles in the kites are uniformly bounded and the radii of one circle pattern converge to $0$. Then a subsequence of the corresponding discrete conformal maps converges to a Riemann map between the given domains.