Discrete harmonic maps and convergence to conformal maps. I: Combinatorial harmonic coordinates. (Q2351853)

Summary: 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 [\textit{S. Hersonsky}, ``Combinatorial Harmonic Maps and Convergence to Conformal Maps. II: Convergence and Moduli'', in preparation], 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 \textit{K. Stephenson} [Rend. Semin. Mat. Fis. Milano 66 (1996), 201--291 (1998; Zbl 0997.60504), Section 11].