Rigidity of Schottky sets

Abstract

We call the complement of a union of at least three disjoint (round) open balls in the unit sphere ${\Bbb S}^n$ a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a M\"obius transformation on ${\Bbb S}^n$. In the other direction we show that every Schottky set in ${\Bbb S}^2$ of positive measure admits nontrivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.