We give the best possible upper bound on the number of exceptional values and
the totally ramified value number of the hyperbolic Gauss map for
pseudo-algebraic constant mean curvature one surfaces in the hyperbolic
three-space and some partial results on the Osserman problem for algebraic
case. Moreover, we study the value distribution of the hyperbolic Gauss map for
complete constant mean curvature one faces in de Sitter three-space.Comment: 16 pages, corrected some typos. OCAMI Preprint Series 08-1, to appear
in Osaka Journal of Mathematic