Addendum: the case of closed surfaces. (Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem)

TL;DR

This paper extends discrete uniformization theorems from planar domains to closed surfaces of non-positive genus, advancing the understanding of boundary value problems on flat surfaces with cone singularities.

Contribution

It generalizes existing uniformization results to closed surfaces, providing new insights into boundary value problems on such geometries.

Findings

Extended uniformization theorems to closed surfaces of non-positive genus

Established boundary value problem solutions on flat surfaces with cone singularities

Connected planar graph results to closed surface cases

Abstract

We extend our discrete uniformization theorems for planar, $m$-connected, Jordan domains [Journal f\"ur die reine und angewandte Mathematik 670 (2012), 65--92] to closed surfaces of non-positive genus.