Quasiconformal homeomorphisms and the convex hull boundary. (English) Zbl 1064.30044

Let \(\Omega\) be an open simply-connected subset of the Riemann sphere \(S^2\) (regarded as the boundary of hyperbolic 3-space \({\mathbf H}^3\)) and let \(X=S^2\setminus \Omega\). (To avoid special cases, we suppose that \(\Omega\subset {\mathbf C}\) and \(\Omega\not= {\mathbf C}\).) We can form the hyperbolic convex hull \({\mathcal CH}(X)\), and the authors denote the relative boundary of \({\mathcal CH}(X)\) in \({\mathbf H}^3\) by \(\text{Dome}(\Omega)\). The study of the geometry of objects such as \(\text{Dome}(\Omega)\) was initiated by Thurston who proved for instance that the hyperboloic metric of \({\mathbf H}^3\) induces a path metric on \(\text{Dome}(\Omega)\) which makes it isometric to the hyperbolic disk \({\mathbf D}^2\). For such an \(\Omega\), Thurston defined a “nearest point retraction” \(r:\Omega\to \text{Dome}(\Omega)\) as follows: for each \(z\in\Omega\), we expand a small horoball at \(z\) and we call \(r(z)\in\text{Dome}(\Omega)\) 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 \(\Omega\not= {\mathbf C}\), there is a \(K\)-quasiconformal homeomorphism \(\Psi: \text{Dome}(\Omega)\to\Omega\) which extends continuously to the identity map on the common boundary \(\partial\Omega\). 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 \(\Omega\). 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 \({\mathbf D}^2\) 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 \(K\).


30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30F60 Teichmüller theory for Riemann surfaces
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
32F17 Other notions of convexity in relation to several complex variables
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