Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics

Abstract

We first present an alternative derivation of a local Weierstrass representation for flat surfaces in the real hyperbolic three-space, $\mathbb{H}^3$, using as a starting point an old result due to Luigi Bianchi. We then prove the following: let $M\subset \mathbb{H}^3$ be a flat compact connected smooth surface with $\partial M\neq \emptyset$, transversal to a foliation of $\mathbb{H}^3$ by horospheres. If, along $\partial M$, $M$ makes a constant angle with the leaves of the foliation, then $M$ is part of an equidistant surface to a geodesic orthogonal to the foliation. We also consider the caustic surface associated with a family of parallel flat surfaces and prove that the caustic of such a familyis also a flat surface (possibly with singularities). Finally, a rigidity result for flat surfaces with singularities and a geometrical application of Schwarz's reflection principle are shown.