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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The Dirichlet problem

1Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2012, Issue 670, Pages 65–92, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2011.155, October 2011

Publication History

Received:
2010-02-08
Published Online:
2011-10-24

Abstract

Consider a planar, bounded, m-connected region Ω, and let Ω be its boundary. Let 𝒯 be a cellular decomposition of Ω ∪ Ω, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S, f) where S is a genus (m − 1) singular flat surface tiled by rectangles and f is an energy preserving mapping from 𝒯(1) onto S. By a singular flat surface, we will mean a surface which carries a metric structure locally modeled on the Euclidean plane, except at a finite number of points. These points have cone singularities, and the cone angle is allowed to take any positive value (see for instance [28] for an excellent survey). Our realization may be considered as a discrete uniformization of planar bounded regions.

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