Rigidity of Schottky sets
Mathematics - Metric Geometry
FOS: Mathematics
Metric Geometry (math.MG)
0101 mathematics
01 natural sciences
DOI:
10.1353/ajm.0.0045
Publication Date:
2009-03-22T09:00:10Z
AUTHORS (3)
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.
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