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

Sa'ar Hersonsky

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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 δΩ be its boundary. Let T denote a triangulation of Ω ∪ δΩ. We construct a new decomposition of Ω ∪ δΩ into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a pair of harmonic functions on T(0) 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 conjecture raised by Stephenson [41, Section 11].

Original languageEnglish
Pages (from-to)325-364
Number of pages40
JournalCommentarii Mathematici Helvetici
Volume90
Issue number2
DOIs
StatePublished - 1 Jan 2015
Externally publishedYes

Keywords

  • Discrete uniformization theorems
  • Flat surfaces with conical singularities
  • Harmonic functions on graphs
  • Planar networks

ASJC Scopus subject areas

  • Mathematics (all)

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