Arkiv för Matematik

Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

Christopher J. Bishop

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Abstract

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.

Note

The author is partially supported by NSF Grant DMS 9800924.

Article information

Source
Ark. Mat., Volume 40, Number 1 (2002), 1-26.

Dates
Received: 4 October 2000
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898750

Digital Object Identifier
doi:10.1007/BF02384499

Mathematical Reviews number (MathSciNet)
MR1948883

Zentralblatt MATH identifier
1034.30013

Rights
2002 © Institut Mittag-Leffler

Citation

Bishop, Christopher J. Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture. Ark. Mat. 40 (2002), no. 1, 1--26. doi:10.1007/BF02384499. https://projecteuclid.org/euclid.afm/1485898750


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