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 $H$ all of whose integral curves are geodesics. Each geodesic foliation of $H$ determines an $n$-dimensional submanifold of the $2n$-dimensional manifold $\mathcal {L}$ of all the oriented geodesics of $H$ (up to orientation preserving reparametrizations). The space $\mathcal {L}$ 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 $\mathcal {L}$ are space-like.