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

TL;DR

This paper introduces a new framework for discrete flat and linear Weingarten surfaces in hyperbolic 3-space, connecting integrable systems with geometric surface theory and analyzing singularities and special examples.

Contribution

It defines discrete flat surfaces in hyperbolic 3-space, relates them to known constant mean curvature surfaces, and explores their singularities and special cases like the Airy equation example.

Findings

Discrete flat surfaces correspond to discrete constant mean curvature 1 surfaces.

Discrete focal surfaces can be used to identify singularities.

An example exhibits swallowtail singularities and Stokes phenomena.

Abstract

We define discrete flat surfaces in hyperbolic 3-space 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 surfaces in hyperbolic 3-space, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. We also examine discrete linear Weingarten surfaces of Bryant type in hyperbolic 3-space, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.