Annales Academię Scientiarum Fennicę
Mathematica
Volumen 36, 2011, 195-214
Yale University, Mathematics Department
10 Hillhouse Ave, New Haven, CT 06520-8283, U.S.A.; mattfeiszli 'at' gmail.com
Abstract. Given a simply connected planar domain \Omega we develop estimates for boundary derivatives on \partial \Omega and estimates for hyperbolic and extremal distances in \Omega and the hyperbolic convex hull boundary S\Omega. 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 applications, 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.
2000 Mathematics Subject Classification: Primary 30C85; Secondary 31A15, 30C35.
Key words: Harmonic measure, extremal length, convex hulls, hyperbolic 3-manifolds, conformal mappings, medial axis.
Reference to this article: M. Feiszli: Extremal distance, hyperbolic distance, and convex hulls over domains with smooth boundary. Ann. Acad. Sci. Fenn. Math. 36 (2011), 195-214.
doi:10.5186/aasfm.2011.3612
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