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

Sa'ar Hersonsky

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arXiv · 1208.2703

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

Sa'ar Hersonsky

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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 $Ω$ and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ denote a triangulation of $Ω\cup\bordΩ$. We construct a \emph{new} decomposition of $Ω\cup\bordΩ$ 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}.

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Sa'ar Hersonsky. 2013-12-22. Combinatorial Harmonic Maps and Convergence to Conformal Maps, I: A Harmonic Conjugate. https://arxiv.org/abs/1208.2703

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