A characterization of Weingarten surfaces in hyperbolic 3-space

TL;DR

This paper explores the geometry of certain surfaces in hyperbolic space, linking properties of geodesic submanifolds to classical surface theory, and classifies special null surfaces with known geometric structures.

Contribution

It characterizes Weingarten surfaces in hyperbolic 3-space via properties of Lagrangian submanifolds in the space of geodesics, providing new classifications and connections to classical surface theory.

Findings

Zero Gauss curvature on Lagrangian surfaces corresponds to Weingarten orthogonal surfaces.

Classified totally null surfaces in the space of geodesics.

Reconstructed known flat and CMC 1 surfaces in hyperbolic 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 K\"ahler 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 ${\mathbb{H}}^3$.