Calibrated geodesic foliations of the hyperbolic space

TL;DR

This paper investigates the geometry of geodesic foliations in hyperbolic space, focusing on volume maximization of associated space-like submanifolds using calibration techniques.

Contribution

It introduces a calibration method to analyze volume maximization for specific geodesic foliations in hyperbolic space.

Findings

Identification of a class of volume-maximizing geodesic foliations

Application of split special Lagrangian calibration in hyperbolic geometry

Characterization of space-like submanifolds in the space of geodesics

Abstract

Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguished geodesic foliations, whose corresponding submanifolds of L are space-like.