Infinite rigidity of inversive distance circle packings in the Poincar\'{e} disk

TL;DR

This paper establishes a maximum principle for hyperbolic inversive distance circle packings, leading to an infinite rigidity theorem for weighted Delaunay triangulations in the Poincaré disk, generalizing previous results.

Contribution

It introduces a unified maximum principle for hyperbolic circle packings and proves an infinite rigidity theorem extending existing hyperbolic rigidity results.

Findings

Established a maximum principle for hyperbolic inversive distance circle packings.

Proved an infinite rigidity theorem for weighted Delaunay triangulations in the Poincaré disk.

Generalized previous hyperbolic rigidity results.

Abstract

The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application of this principle, a discrete Schwarz-Ahlfors lemma is established.Furthermore, an infinite rigidity theorem for weighted Delaunay triangulations of the Poincar\'{e} disk is proved,which generalizes He's hyperbolic rigidity result \cite{He2}.