A characterization of Weingarten surfaces in hyperbolic 3-space

Abstract

We study 2-dimensional submanifolds of the space $\mathbb{L}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ${\mathbb{H}}^3$ orthogonal to the geodesics of $\Sigma$. We prove that the induced metric on a Lagrangian surface in $\mathbb{L}({\mathbb{H}}^3)$ has zero Gauss curvature iff the orthogonal surfaces in ${\mathbb{H}}^3$ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in $\mathbb{L}({\mathbb{H}}^3)$ and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ${\mathbb{H}}^3$.

20 pages AMS-LATEX, version 2 correction of typos, inclusion of non-graph cases

References in corpus (2)

• Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic 3-Space
• Totally Null Surfaces in Neutral Kaehler 4-Manifolds

Cited by in corpus (7)

• Marginally trapped surfaces in spaces of oriented geodesics
• Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces
• Totally Null Surfaces in Neutral Kaehler 4-Manifolds
• From CT scans to 4-manifold topology
• Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space
• On the Convergence of Non-Integer Linear Hopf Flow
• Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces