2012/08/13 by Sa’ar Hersonsky, Hersonsky, Sa'ar
Mathematics · #30G25 #39A12 #53C43 #57M50 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1208.2703
openalex publication_date 2012/08/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 T denote a triangulation of Ω∪\bordΩ. We construct a new decomposition of Ω∪\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 \citeHer3 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.