2015/11/23 by Younsi, Malik · 2 citations
#30C35 #30F40 #Complex Variables (math.CV) #FOS: Mathematics #primary 30C20 #secondary 30C62
paper · doi:10.48550/arxiv.1511.07348
A circle domain Ω in the Riemann sphere is conformally rigid if every conformal map of Ω onto another circle domain is the restriction of a Möbius transformation. We show that two rigidity conjectures of He and Schramm are in fact equivalent, at least for a large family of circle domains. The proof follows from a result on the removability of countable unions of certain conformally removable sets. We also introduce trans-quasiconformal deformation of Schottky groups to prove that a circle domain is conformally rigid if and only if it is quasiconformally rigid, thereby providing new evidence for the aforementioned conjectures.