Applications of hyperbolic convexity to Euclidean and spherical convexity
The author gives a number of applications of a reflection principle for the hyperbolic metric on Riemann surfaces that he established [Complex Variables 8, 129-144 (1987; Zbl 0576.30022)]. For the purposes of this paper the following geometric consequence of the reflection principle is essential: if \(\Omega\) is a hyperbolic region in \({\mathbb{C}}\) and \(\Omega\) is starlike with respect to the point a, then \(\Omega\cap \{z:| z- a| <r\}\) is a hyperbolically convex subset of \(\Omega\) for any \(r>0\). This sufficient condition for hyperbolic convexity has applications (which are sharp) to regions in \({\mathbb{C}}\) that are convex relative to Euclidean geometry and to regions on the Riemann sphere \({\mathbb{P}}\) that are convex relative to spherical geometry. For instance, if \(\gamma\) is a hyperbolic geodesic in a convex region \(\Omega\) in \({\mathbb{C}}\) and \(z_ 0\in \gamma\), then the center of the Euclidean circle of curvature for \(\gamma\) at \(z_ 0\) lies in the complement of \(\Omega\) ; moreover, the center is on the boundary of \(\Omega\) if and only if \(\Omega\) is either a half-plane or a sector with angular opening strictly less than \(\pi\). This geometric result implies a number of distortion theorems for the hyperbolic metric in a convex region and for normalized convex univalent functions. Analogous results hold for regions that are spherically convex.
scientific article; zbMATH DE number 4040232
scientific article