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We extend our discrete uniformization theorems for planar,  m -connected, Jordan domains [J. reine angew. Math. 670 (2012), 65–92] to closed surfaces of genus  m  ≥ 1.

journal für die reine und angewandte mathematik (crelles journal)

Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)