An improved bound for Sullivan's convex hull theorem (Q2795905)

Let \(\Omega\) be a hyperbolic simply connected domain in the extended complex plane (\( \partial H^3\)) and \(\mathrm{Dome}(\Omega)\) the hyperbolically convex hull of its complement. \textit{D. Sullivan} [Lect. Notes Math. 842, 196--214 (1981; Zbl 0459.57006)] showed that there exists a \(K_0\)-quasiconformal map \(f : \Omega \rightarrow \mathrm{Dome}(\Omega)\) which extends to the identity on \(\partial \Omega\). Moreover, \(f\) is natural in the sense that if \(A\) is a conformal automorphism of \(\mathbb{C}\) preserving \(\Omega\), then \(A' \circ f = f \circ A\) where \(A'\) is the extension of \(A\) to an isometry of \(H^3\). Upper and lower bounds for \(K_0\) are due to \textit{D. B. A. Epstein} et al. [Ann. Math. (2) 159, No. 1, 305--336 (2004; Zbl 1064.30044); J. Differ. Geom. 73, No. 1, 119--166 (2006; Zbl 1114.30047)] who showed that \(2.1 < K_0 \leq 13.88\) and later by \textit{C. J. Bishop} [Contemp. Math. 355, 41--69 (2004; Zbl 1069.30030)] who showed that \(K_0 \leq 7.88\) if one does not require that \(f\) is conformally natural. The authors improve these bounds. They first improve the bounds of the first author [Mich. Math. J. 51, No. 2, 363--378 (2003; Zbl 1065.30041)] on ``roundness'' which measures the total bending of the realization of \(\mathrm{Dome}(\Omega)\) as a pleated plane \(P_{\mu} :H^2 \rightarrow H^3\) and then they improve the work of Epstein et al. [loc. cit.] to get criteria for \(P_{\mu}\) to be an embedding. These considerations lead to \(K_0 \leq 7.1695\).