Hyperbolic polygons and simple closed geodesics
In this paper the author studies a universal constant for simple closed geodesics and the radius of a disk on closed hyperbolic Riemann surfaces by using the properties of the right-angled hyperbolic polygons. Let \(S\) be a closed Riemann surface of genus \(g>1\) with a metric of constant curvature \(-1\). Then it is proved that for every simple closed geodesic \(\gamma\) on \(S\) there exist \(4g-4\) closed disks of radius \(\frac 1 2 \ln 3\) contained in \(S\setminus \gamma\). Furthermore, if \(\rho>\frac 1 2 \ln 3\), then there exists a simple closed geodesic \(\gamma_{\rho}\) such that \(S\setminus \gamma_{\rho}\) does not contain any open disks of radius \(\rho\). The author states the result for hyperbolic Riemann surfaces of signature \((g,n)\) and Klein surfaces with respect to Marden's universal constant for Fuchsian groups.
scientific article; zbMATH DE number 5141212