Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem

TL;DR

This paper introduces a method to solve boundary value problems on planar graphs by constructing singular flat surfaces with cone points, linking discrete graph data to continuous geometric structures.

Contribution

It presents a novel construction of singular flat surfaces from cellular decompositions and conductance functions, providing a geometric framework for boundary value problems.

Findings

Constructs a canonical pair (S,f) from graph data and conductance.

Creates singular flat surfaces tiled by rectangles with cone singularities.

Establishes a mapping preserving energy between graph edges and the surface.

Abstract

Consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$, 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 ${\mathcal T}^{(1)}$ onto $S$.