Rigidity of Schottky sets
  We call a complement of a union of at least three disjoint (round) open balls in the unit sphere 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 Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial 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. 
Mario Bonk, Bruce Kleiner, Sergei Merenkov
math.MG