Calibrated geodesic foliations of hyperbolic space

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 L all of whose integral curves are geodesics. Each geodesic foliation of L determines an n-dimensional submanifold 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 L. 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.