Discrete Conformal Geometry of Polyhedral Surfaces and Its Convergence

TL;DR

This paper demonstrates that discrete conformal maps of polyhedral surfaces converge to the classical Riemann mapping, extending circle packing results to a new setting with rigorous convergence proofs.

Contribution

It introduces a convergence theorem for discrete conformal maps of polyhedral surfaces to Riemann mappings, paralleling circle packing theorems with a novel approach.

Findings

Discrete conformal maps converge to Riemann mappings for Jordan domains.

Established a rigidity result for regular hexagonal triangulations.

Provided estimates for quasiconformal constants in discrete conformality.

Abstract

The paper proves a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivan's theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of discrete conformality. The proof follows the same strategy that Rodin-Sullivan used by establishing a rigidity result for regular hexagonal triangulations of the plane and estimating the quasiconformal constants associated to the discrete conformal maps.