The main goal of this paper is to reveal the geometric meaning of the maximal
number of exceptional values of Gauss maps for several classes of immersed
surfaces in space forms, for example, complete minimal surfaces in the
Euclidean three-space, weakly complete improper affine spheres in the affine
three-space and weakly complete flat surfaces in the hyperbolic three-space.
For this purpose, we give an effective curvature bound for a specified
conformal metric on an open Riemann surface.Comment: 13 pages, to appear in Mathematische Zeitschrif