Hyperbolic geometry and disks
The paper contains two characterizations for a Euclidean disk in terms of hyperbolic geometry. In order to state the first one we need a couple of simple definitions. A plane domain \(G\) is circularly accessible at a point \(w\in\partial G\) if there exists a disk \(B_w\subset G\) with \(w\in \partial B_w\). For a bounded domain \(G\) the points \(w_1,w_2\) are diametral points if \(w_1,w_2\in \partial G\) and \(|w_1-w_2|=\text{diam}(G)\). The first characterization is: A bounded simply connected domain \(D\) is a disk if and only if \(D\) is circularly accessible at a pair of diametral points and \(h_D(z_1,z_2)\leq j_D(z_1,z_2)\), for \(z_1,z_2\in D\). Here \(h_D\) denotes the hyperbolic distance in \(D\) and \(j_D\) is another metric in \(D\) that involves only Euclidean quantities (distances between \(z_1, z_2\) and \(\partial D\)). The second characterization involves convexity. A simply connected domain \(D\subset \bar{\mathbb C}\) with \(\bar{D}\neq \bar{\mathbb C}\) is a disk or half-plane if it is a hyperbolically convex subset of each simply connected domain which contains it. This result was originally proved by \textit{B. Brown Flinn} [Indiana Univ. Math. J. 32, 831-841 (1983; 522.30005)].
scientific article; zbMATH DE number 1334869
scientific article