Rigidity of configurations of balls and points in the $N$-sphere

TL;DR

This paper establishes that configurations of balls and points in the N-sphere are uniquely determined by signed inversive distances and cross-ratios, respectively, except in specific boundary cases, using hyperboloid models of hyperbolic space.

Contribution

It provides a complete characterization of configurations in the N-sphere via geometric invariants, answering open questions in conformal geometry.

Findings

Configurations of balls are determined by signed inversive distances, except when boundaries intersect.

Configurations of points are determined by absolute cross-ratios of quadruples.

The proofs utilize the hyperboloid model of hyperbolic space.

Abstract

We answer two questions of Beardon and Minda that arose from their study of the conformal symmetries of circular regions in the complex plane. We show that a configuration of closed balls in the $N$-sphere is determined up to M\"{o}bius transformations by the signed inversive distances between pairs of its elements, except when the boundaries of the balls have a point in common, and that a configuration of points in the $N$-sphere is determined by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic $(N+1)$-space.