Convergence of discrete conformal mappings on surfaces

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.