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

### Francisco Martín

Universidad de Granada, Spain### Masaaki Umehara

Tokyo Institute of Technology, Japan### Kotaro Yamada

Tokyo Institute of Technology, Japan

## 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).

## Cite this article

Francisco Martín, Masaaki Umehara, Kotaro Yamada, Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 309–316

DOI 10.4171/RMI/779