Discrete uniformization of polyhedral surfaces

Abstract

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.

arXiv comment45 pages, 27 figures, comments are welcome. Revised version. Added an appendix containing the deferred proof of Lemma 2.7, expanded several proofs, and corrected minor errors