Convergence of discrete conformal mappings on surfaces

TL;DR

This paper proves a general convergence theorem for discrete conformal mappings on surfaces, encompassing various structures and ensuring they approximate true conformal maps under certain conditions.

Contribution

It introduces a unified convergence theorem for discrete conformal maps on surfaces, extending previous results to broader structures and manifolds.

Findings

Convergence of barycentric discrete conformal maps to true conformal maps.

Estimates for the pullback of Riemannian metrics ensure conformality.

Conditions on simplex fullness prevent degenerate triangles.

Abstract

Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence result of Rodin-Sullivan (1987) proved that circle packing maps do indeed converge to conformal maps to the disk. Recent results have shown convergence of maps of other discrete conformal structures to conformal maps as well. We give a general theorem of convergence of discrete conformal mappings between surfaces that allows for a variety of discrete conformal structures and manifolds with or without boundary. The mappings are a composition of piecewise linear discrete conformal mappings and Riemannian barycentric coordinates, called barycentric discrete conformal maps. Estimates of the barycentric discrete conformal maps allow extraction of convergent subsequences and estimates for the pullback of the Riemannian metric, proving conformality. The theorem requires assumptions on fullness of simplices to prevent degenerate triangles and a local discrete conformal rigidity generalizing hexagonal rigidity of circle packings.