We elucidate the geometric background of function-theoretic properties for
the Gauss maps of several classes of immersed surfaces in three-dimensional
space forms, for example, minimal surfaces in Euclidean three-space, improper
affine spheres in the affine three-space, and constant mean curvature one
surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose,
we prove an optimal curvature bound for a specified conformal metric on an open
Riemann surface and give some applications. We also provide unicity theorems
for the Gauss maps of these classes of surfaces.Comment: 26 pages. arXiv admin note: substantial text overlap with
arXiv:1205.478