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

  • Sa'ar Hersonsky

    University of Georgia, Athens, USA

Abstract

In this paper, we provide new discrete uniformization theorems for bounded, -connected planar domains. To this end, we consider a planar, bounded, -connected domain , and let be its boundary. Let denote a triangulation of . We construct a new decomposition of into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a pair of harmonic functions on 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].

Cite this article

Sa'ar Hersonsky, Discrete harmonic maps and convergence to conformal maps, I: Combinatorial harmonic coordinates. Comment. Math. Helv. 90 (2015), no. 2, pp. 325–364

DOI 10.4171/CMH/355