Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant (Q1041552)

The authors consider D. Sullivan's dome construction and quasiconformal maps, previously studied also by \textit{D. B. A. Epstein} and \textit{A. Marden} [in: Analytical and geometric aspects of hyperbolic space, Symp. Warwick and Durham/Engl. 1984, Lond. Math. Soc. Lect. Note Ser. 111, 113-253 (1987; Zbl 0612.57010)], and more recently by \textit{C. J. Bishop} [Ark. Mat. 40, No. 1, 1--26 (2002; Zbl 1034.30013)] and \textit{A. Marden} and \textit{V. Markovic} [Bull London Math Soc 39, 962--972 (2007; Zbl 1145.30008)]. For a multiply connected domain \(\Omega \) in \(\mathbb R^{2}\), let \(S\) be the boundary of the convex hull in \(H^{3}\) of \(\mathbb R^{2}\setminus \Omega \) which faces \(\Omega \). Suppose in addition that there exists a lower bound \(l > 0\) of the hyperbolic lengths of the closed geodesics in \(\Omega \). The authors prove that there is a \(K\)-quasiconformal mapping from \(S\) to \(\Omega \), which extends continuously to the identity on \(\partial S = \partial \Omega \), where \(K\) depends only on \(l\). The authors also give a numerical estimate of \(K\) by using the parameter \(l\). The paper is closely related to the work of Marden and Markovic.