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The logarithmic spiral: a counterexample to the K = 2 conjecture

Epstein, D. B. A. and Markovic, V. (2005) The logarithmic spiral: a counterexample to the K = 2 conjecture. Annals of Mathematics, 161 (2). pp. 925-957. ISSN 0003-486X.

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Given a nonempty compact connected subset X ⊂ S^2 with complement a simply-connected open subset Ω ⊂ S^2, let Dome(Ω) be the boundary of the hyperbolic convex hull in H^3 of X. We show that if X is a certain logarithmic spiral, then we obtain a counterexample to the conjecture of Thurston and Sullivan that there is a 2-quasiconformal homeomorphism Ω → Dome(Ω) which extends to the identity map on their common boundary in S^2. This leads to related counterexamples when the boundary is real analytic, or a finite union of intervals (straight intervals, if we take S^2 = C ∪ {∞}. We also show how this counterexample enables us to construct a related counterexample which is a domain of discontinuity of a torsion-free quasifuchsian group with compact quotient. Another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately .98π/2, which is substantially larger than that of any previously known example.

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Additional Information:© 2005 Annals of Mathematics. Received: 9 January 2002 Revised: 5 March 2003 Accepted: 12 February 2003 Published online: 1 March 2009
Issue or Number:2
Record Number:CaltechAUTHORS:20170509-065233646
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77281
Deposited By: Ruth Sustaita
Deposited On:09 May 2017 20:56
Last Modified:03 Oct 2019 17:55

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