  Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a
closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We
construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes
whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are
deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic
planes projected down from the universal cover $\mathbb{H}^3$. Our construction
continues to work as long as the sum of the squares of the principal curvatures
of the (projections to $M$) of the leaves remains pointwise smaller in
magnitude than the ambient Ricci curvature in the normal direction. In the
second part of the paper we give some applications and construct negatively
curved metrics for which $Gr_2(M)$ cannot admit a foliation as above.
