2011/09/21 by Sergei Merenkov · 1 citation
Mathematics · #Analytic and geometric function theory #Geometry and complex manifolds #Holomorphic and Operator Theory #math.MG
paper · pdf · doi:10.1112/plms/pdr038
published as Proc. Lond. Math. Soc. (3) 104 (2012), 455-485 · 43 pages
openalex publication_date 2011/09/21 · arxiv created 2013/05/17 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A relative Schottky set in a planar domain Ω is a subset of Ω obtained by removing from Ω open geometric disks whose closures are in Ω and are pairwise disjoint. In this paper, we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasisymmetric maps for relative Schottky sets in the unit disk.