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

TL;DR

This paper constructs weakly complete flat surfaces in hyperbolic 3-space with both hyperbolic Gauss maps bounded within a specified boundary, revealing a phenomenon contrasting minimal surface theory in Euclidean and hyperbolic spaces.

Contribution

It introduces a method to construct such surfaces using minimal surface theory, demonstrating the existence of bounded hyperbolic Gauss maps in hyperbolic space.

Findings

Existence of weakly complete flat surfaces with bounded hyperbolic Gauss maps

Construction method based on minimal surface theory

Contrasts with non-existence results in Euclidean and hyperbolic spaces

Abstract

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as an application of the minimal surface theory. This looks an interesting phenomenon if one comparing the fact that there are no complete minimal (resp. constant mean curvature one) surfaces in R^3 (resp. H^3) having bounded Gauss maps (resp. bounded hyperbolic Gauss maps).