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Quasiconformal homeomorphisms and the convex hull boundary. (English) Zbl 1064.30044
Let Ω be an open simply-connected subset of the Riemann sphere S 2 (regarded as the boundary of hyperbolic 3-space 𝐇 3 ) and let X=S 2 Ω. (To avoid special cases, we suppose that Ω𝐂 and Ω𝐂.) We can form the hyperbolic convex hull 𝒞H(X), and the authors denote the relative boundary of 𝒞H(X) in 𝐇 3 by Dome(Ω). The study of the geometry of objects such as Dome(Ω) was initiated by Thurston who proved for instance that the hyperboloic metric of 𝐇 3 induces a path metric on Dome(Ω) which makes it isometric to the hyperbolic disk 𝐃 2 . For such an Ω, Thurston defined a “nearest point retraction” r:ΩDome(Ω) as follows: for each zΩ, we expand a small horoball at z and we call r(z)Dome(Ω) the unique first point of contact. Sullivan, and Epstein-Marden analysed that construction and they proved that there exists K such that for any simply connected Ω𝐂, there is a K-quasiconformal homeomorphism Ψ:Dome(Ω)Ω which extends continuously to the identity map on the common boundary Ω. Thurston suggested that the best constant K is 2, and this suggestion has been called later on “Thurston’s K=2 conjecture”. In this paper, the authors give a counterexample to that conjecture in its equivariant form, that is, in the case where the homeomorphism respects a group of Möbius transformations which preserve Ω. Another result that the authors prove in this paper is that the nearest point retraction r is 2-Lipschitz in the respective hyperbolic metrics, and that the constant 2 here is best possible. The authors also study pleating maps of hyperbolic 2-plane. They obtain explicit universal constants 0<c 1 <c 2 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. They show that every K-quasiconformal homeomrophism of the unit disk 𝐃 2 is a (K,a(K))-quasi-isometry, where a(K) is an explicitely computed function, where the multiplicative constant is best possible and where the additive constant a(K) is best possible for some values of K.
MSC:
30F45Conformal metrics (hyperbolic, Poincaré, distance functions)
30F60Teichmüller theory
37F30Quasiconformal methods and Teichmüller theory, etc. (dynamical systems)
57M50Geometric structures on low-dimensional manifolds
32F17Other notions of convexity