Tohoku Mathematical Journal
2007
March
SECOND SERIES VOL. 59, NO. 1

Tohoku Math. J.
59 (2007), 21-37

Title FLAT SURFACES IN HYPERBOLIC SPACE AS NORMAL SURFACES TO A CONGRUENCE OF GEODESICS

Author Pedro Roitman

(Received July 19, 2005, revised June 22, 2006)
Abstract. We first present an alternative derivation of a local Weierstrass representation for flat surfaces in the real hyperbolic three-space, $\boldsymbol{H}^3$, using as a starting point an old result due to Luigi Bianchi. We then prove the following: let $M \subset \boldsymbol{H}^3$ be a flat compact connected smooth surface with $\partial M \neq \emptyset$, transversal to a foliation of $\boldsymbol{H}^3$ 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 family is 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.

2000 Mathematics Subject Classification. Primary 53A10; Secondary 53C24, 53C42.

Key words and phrases. Flat surfaces, caustics, Weierstrass representation.

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