## 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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