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

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.