Open Access
April 2002 Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture
Christopher J. Bishop
Author Affiliations +
Ark. Mat. 40(1): 1-26 (April 2002). DOI: 10.1007/BF02384499


We show that proving the conjectured sharp constant in a theorem of Dennis Sullivan concerning convex sets in hyperbolic 3-space would imply the Brennan conjecture. We also prove that any conformal map f: D→Ω can be factored as a K-quasiconformal self-map of the disk (withK independent of Ω) and a map g: D→Ω with derivative bounded away from zero. In particular, there is always a Lipschitz homeomorphism from any simply connected Ω (with its internal path metric) to the unit disk.

Funding Statement

The author is partially supported by NSF Grant DMS 9800924.


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Christopher J. Bishop. "Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture." Ark. Mat. 40 (1) 1 - 26, April 2002.


Received: 4 October 2000; Published: April 2002
First available in Project Euclid: 31 January 2017

zbMATH: 1034.30013
MathSciNet: MR1948883
Digital Object Identifier: 10.1007/BF02384499

Rights: 2002 © Institut Mittag-Leffler

Vol.40 • No. 1 • April 2002
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