Discrete harmonic maps and convergence to conformal maps, I: Combinatorial harmonic coordinates

Abstract

In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $\Omega$, and let $\partial \Omega$ be its boundary. Let $\mathcal{T}$ denote a triangulation of $\Omega \cup \partial \Omega$. We construct a new decomposition of $\Omega \cup \partial \Omega$ into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a pair of harmonic functions on ${\mathcal{T}}^{(0)}$ and properties of their level curves. In the sequel [26], it will be proved that a particular discrete scheme based on these theorems converges to a conformal map, thus providing an affirmative answer to a question raised by Stephenson [41, Section 11].