Extremal distance, hyperbolic distance, and convex hulls over domains with smooth boundary

Abstract

Given a simply connected planar domain › we develop estimates for boundary derivatives on @› and estimates for hyperbolic and extremal distances in › and the hyperbolic convex hull boundary S›. We focus on the case when the underlying domain has smooth boundary; this allows very explicit formulas in terms of a collection of invariants which clarify behavior even in the generic case. In particular, we are able to obtain very explicit estimates using the intimate connection between the convex hull boundary and the geometry of the medial axis. As applica- tions, we include here a refinement and alternate proof of the Thurston-Sullivan conjecture that the nearest-point retraction is 2-Lipschitz in the hyperbolic metrics and a variant of the Ahlfors distortion theorem which works as an integral along branches of the medial axis. In this paper, we study the special case of the hyperbolic convex hull boundary S› over a domain › ‰ C with smooth boundary; in particular we obtain a number of estimates and invariants which relate extremal lengths and hyperbolic distances upstairs in the convex hull boundary to the same quantities downstairs in the domain via the nearest-point retraction map r: › ! S›. The study of the smooth case was originally motivated, perhaps unexpectedly, by computer vision, where con- formal mapping techniques were introduced by Sharon and Mumford (see (25)) to construct metric spaces of smooth curves. In this application, smooth Jordan curves are represented by their associated welding maps, and a Riemannian metric on curves is induced by the Weil-Peterson metric on the dieomorphism group of the circle. The curves of interest are typically at least C 2 , and careful geometric estimates for boundary derivatives and extremal distances are required in order to understand geodesic distances between curves. We obtain these by working with the convex hull boundary S›. The surface S› admits an explicit construction for the Riemann map ¶: S› ! D (see, for example, (6) or (18)) and this mapcan act as a sort of poor-man's Riemann map from the disk to › itself. Among the advantages is that S› is a ruled surface, and this special geometry permits easier analysis; in addition, the Riemann mapis an entirely local construction, unlike the global nature of the Riemann map to ›. The resulting formulas are, for the most part, simple and explicit, and we feel that working in the smooth case has the potential to simplify many computations