Maximal principles in discrete conformal geometry with application to the rigidity of infinite triangulations

TL;DR

This paper develops maximum principles for discrete conformal structures on polyhedral surfaces, unifying existing theories and applying them to prove a Schwarz-Ahlfors lemma and an infinite rigidity theorem for hyperbolic triangulations.

Contribution

It introduces generalized maximum principles for discrete conformal geometry, extending previous results and applying them to new rigidity and mapping theorems in hyperbolic geometry.

Findings

Established maximum principles for Euclidean and hyperbolic discrete conformal structures.

Proved a discrete Schwarz-Ahlfors lemma in hyperbolic geometry.

Demonstrated an infinite rigidity theorem for small Delaunay triangulations.

Abstract

In this paper, maximum principles for Euclidean and hyperbolic discrete conformal structures on polyhedral surfaces are established. These maximum principles unify and generalize the maximum principles for vertex scalings and different types of circle packings in the literature. As an application of the hyperbolic discrete maximum principle, a discrete Schwarz-Ahlfors lemma is established. As another application, an infinite rigidity theorem for small Delaunay triangulations of the hyperbolic plane is proved.