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Quasiconformal homeomorphisms and the convex hull boundary

Epstein, D. B. A. and Marden, A. and Markovic, V. (2004) Quasiconformal homeomorphisms and the convex hull boundary. Annals of Mathematics, 159 (1). pp. 305-336. ISSN 0003-486X. doi:10.4007/annals.2004.159.305.

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We investigate the relationship between an open simply-connected region Ω ⊂ S^2 and the boundary Y of the hyperbolic convex hull in H^3 of S^2∖Ω. A counterexample is given to Thurston’s conjecture that these spaces are related by a 2-quasiconformal homeomorphism which extends to the identity map on their common boundary, in the case when the homeomorphism is required to respect any group of Möbius transformations which preserves Ω. We show that the best possible universal lipschitz constant for the nearest point retraction r:Ω → Y is 2. We find explicit universal constants 0 < c_2 < c_1, such that no pleating map which bends more than c_1 in some interval of unit length is an embedding, and such that any pleating map which bends less than c_2 in each interval of unit length is embedded. We show that every K-quasiconformal homeomorphism D^2 → D^2 is a (K,ɑ(K))-quasi-isometry, where ɑ(K) is an explicitly computed function. The multiplicative constant is best possible and the additive constant ɑ(K) is best possible for some values of K.

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Additional Information:© 2004 Annals of Mathematics. Received June 20, 2001. Accepted: 20 August 2002 Published online: January 2004
Issue or Number:1
Record Number:CaltechAUTHORS:20170508-093038545
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Official Citation:Quasiconformal homeomorphisms and the convex hull boundary Pages 305-336 from Volume 159 (2004), Issue 1 by David B. A. Epstein, Albert Marden, Vladimir Markovic
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77252
Deposited By: Ruth Sustaita
Deposited On:09 May 2017 22:53
Last Modified:15 Nov 2021 17:29

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