Euclidean formulation of discrete uniformization of the disk

TL;DR

This paper reformulates Thurston's circle packing approximation of the Riemann Mapping into a Euclidean framework, enabling more general discrete conformal structures and boundary conditions for disk uniformization.

Contribution

It introduces a Euclidean formulation of discrete uniformization, replacing hyperbolic methods with a model involving double covered disks and zero curvature structures.

Findings

Reformulation allows broader discrete conformal structures.

Zero curvature structures are central to the Euclidean approach.

Curvature measures extend to generalized manifolds.

Abstract

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to the circle. The main proofs of the uniformization use hyperbolic volumes (Andreev) or hyperbolic circle packings (by Beardon and Stephenson). We reformulate these problems into a Euclidean context, which allows more general discrete conformal structures and boundary conditions. The main idea is to replace the disk with a double covered disk with one side forced to be a circle and the other forced to have interior curvature zero. The entire problem is reduced to finding a zero curvature structure. We also show that these curvatures arise naturally as curvature measures on generalized manifolds (manifolds with multiplicity) that extend the usual discrete Lipschitz-Killing curvatures on surfaces.