## 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.

Original language | English |
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Pages (from-to) | 65-92 |

Number of pages | 28 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 670 |

DOIs | |

State | Published - 1 Sep 2012 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics