Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded

Abstract

We construct a weakly complete flat surface in hyperbolic 3-space $H^3$ having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disk in the ideal boundary of $H^3$. This construction is accomplished as an application of minimal surface theory. This is an interesting phenomenon when one compares it with the fact that there are no complete non-flat minimal (resp. non-horospherical constant mean curvature one) surfaces in ${\mathbb{R}}^3$ (resp. $H^3$) having bounded Gauss maps (resp. bounded hyperbolic Gauss maps).