Rigidity of Circle Packings with Crosscuts

TL;DR

This paper explores the rigidity and uniqueness of circle packings with specified tangency patterns within bounded domains, introducing crosscuts and establishing a discrete analogue of the identity theorem for analytic functions.

Contribution

It introduces the concept of crosscuts in circle packings and proves a discrete identity theorem, advancing understanding of discrete conformal mappings.

Findings

Rigidity results for circle packings with maximal crosscuts.

A discrete analogue of Schwarz's Lemma for circle packings.

Implications for uniqueness in discrete conformal mappings.

Abstract

Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a bounded, simply connected domain. We introduce the concept of crosscuts and investigate the rigidity of circle packings with respect to maximal crosscuts. The main result is a discrete version of an indentity theorem for analytic functions (in the spirit of Schwarz' Lemma), which has implications to uniqueness statements for discrete conformal mappings.