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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Schedule a callMore From: Journal für die reine und angewandte Mathematik (Crelles Journal)
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