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

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.