Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space

Abstract

We define discrete flat surfaces in hyperbolic 3 3 -space H 3 \mathbb {H}^3 from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 1 surfaces in H 3 \mathbb {H}^3 , and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in H 3 \mathbb {H}^3 , and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.