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

• 成果类型: Article
• 署名作者: Epstein, DBA; Markovic, V
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.161.925
• 发表日期: 2005
• 页码: 925-957
• 关键词: convex-hull theorem; BOUNDARY; mappings

摘要

Given a nonempty compact connected subset X subset of S-2 with complement a simply-connected open subset Omega subset of S-2, let Dome (Omega) 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 Omega -> Dome (Omega) 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 boolean OR {infinity}). 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 .98 pi/2, which is substantially larger than that of any previously known example.