Tohoku Mathematical Journal

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

Pedro Roitman

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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.

Article information

Source
Tohoku Math. J. (2), Volume 59, Number 1 (2007), 21-37.

Dates
First available in Project Euclid: 16 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1176734745

Digital Object Identifier
doi:10.2748/tmj/1176734745

Mathematical Reviews number (MathSciNet)
MR2321990

Zentralblatt MATH identifier
1140.53004

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53C24: Rigidity results 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
Flat surfaces caustics Weierstrass representation

Citation

Roitman, Pedro. Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics. Tohoku Math. J. (2) 59 (2007), no. 1, 21--37. doi:10.2748/tmj/1176734745. https://projecteuclid.org/euclid.tmj/1176734745


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References

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