On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space

TL;DR

This paper investigates maximal surfaces in the space of oriented geodesics of hyperbolic 3-space, proving holomorphic curves are maximal and classifying Lagrangian maximal surfaces as equidistant tubes around a geodesic.

Contribution

It establishes that all holomorphic curves in this space are maximal surfaces and classifies Lagrangian maximal surfaces as equidistant tubes around a geodesic.

Findings

Holomorphic curves are maximal surfaces in the space of geodesics.

Lagrangian maximal surfaces correspond to equidistant tubes around a geodesic.

Classification of maximal surfaces in the space of oriented geodesics.

Abstract

We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. We prove that every holomorphic curve in ${\mathbb L}({\mathbb H}^3)$ is a maximal surface. We then classify Lagrangian maximal surfaces $\Sigma$ in ${\mathbb L}({\mathbb H}^3)$ and prove that the family of parallel surfaces in ${\mathbb H}^3$ orthogonal to the geodesics $\gamma\in\Sigma$ form a family of equidistant tubes around a geodesic.