Combinatorial Harmonic Maps and Convergence to Conformal Maps, I: A Harmonic Conjugate

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 $\bord\Omega$ be its boundary. Let $\mathcal{T}$ denote a triangulation of $\Omega\cup\bord\Omega$. We construct a \emph{new} decomposition of $\Omega\cup\bord\Omega$ into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a {\it pair} of harmonic functions on ${\mathcal T}^{(0)}$ and properties of their level curves. In the sequel \cite{Her3} 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 \cite[Section 11]{Steph}.