# Revista Matemática Iberoamericana

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**Volume 30, Issue 1, 2014, pp. 309–316**

**DOI: 10.4171/RMI/779**

Published online: 2014-03-23

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

Francisco Martín^{[1]}, Masaaki Umehara

^{[2]}and Kotaro Yamada

^{[3]}(1) Universidad de Granada, Spain

(2) Tokyo Institute of Technology, Japan

(3) Tokyo Institute of Technology, Japan

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).

*Keywords: *Contact manifolds, minimal surfaces, flat fronts, improper affine fronts

Martín Francisco, Umehara Masaaki, Yamada Kotaro: Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded. *Rev. Mat. Iberoamericana* 30 (2014), 309-316. doi: 10.4171/RMI/779