Uniformly perfect domains and convex hulls: Improved bounds in a generalization of a theorem of Sullivan

Abstract

Given a hyperbolic domain Ω, the nearest point retraction is a conformally natural homotopy equivalence from Ω to the boundary Dome(Ω) of the convex core of its complement. Marden and Markovic showed that if Ω is uniformly perfect, then there exists a conformally natural quasiconformal map from Ω to Dome(Ω) which admits a bounded homotopy to the nearest point retraction. We obtain an explicit upper bound on the quasiconformal dilatation which depends only on the injectivity radius of the domain.