  Every nonconstant meromorphic function in the plane univalently covers
spherical discs of radii arbitrarily close to arctan(sqrt 8) ~ 70^\circ 32'. If
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.
