## Abstract

We answer two questions of Beardon and Minda which 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 Mbius 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 up to Mbius transformations by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic (N 1)-space.

Translated title of the contribution | Rigidity of coonfigurations of balls and points in the N-sphere |
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Original language | English |

Pages (from-to) | 351 - 362 |

Number of pages | 12 |

Journal | Quarterly Journal of Mathematics |

Volume | 62 |

Issue number | 2 |

Early online date | 29 Jan 2010 |

DOIs | |

Publication status | Published - Jun 2011 |