HARMONIC MEASURE, EXTREMAL DISTANCE, AND HYPERBOLIC METRICS IN SMOOTH DOMAINS

Given a simply connected planar domain Ω we develop estimates for boundary derivatives on∂ Ω and estimates for hyperbolic and extremal distances in Ω and the hyperbolic convex hull boundary SΩ. We are able to obtain very explicit estimates using the geometry of the medial axis. Results include a refinement and proof of the Thurston-Sullivan conjecture that the nearest-point retraction is 2-Lipschitz in the hyperbolic metrics.