Rigidity of schottky sets

Mario Bonk, Bruce Kleiner, Sergei Merenkov

Research output: Contribution to journalArticlepeer-review


We call the 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 Möbius transformation on S n. In the other direction we show that every Schottky set in 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.

Original languageEnglish (US)
Pages (from-to)409-443
Number of pages35
JournalAmerican Journal of Mathematics
Issue number2
StatePublished - 2009

ASJC Scopus subject areas

  • General Mathematics


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