Abstract
Every nonconstant meromorphic function in the plane univalently covers spherical discs of radii arbitrarily close to arctan $\sqrt{8} \approx 70^{\circ}32’$. In in addition all critical points of the function are multiple, then a similar statement holds with $\pi/2$. These constants are the best possible. The proof is based on the consideration of negatively curved singular surfaces associated with meromorphic functions.
