Singular Monge-Ampere foliations

This paper generalizes results of Lempert and Szoke on, the structure of the singular set of a solution of the homogeneous Monge-Ampere equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C-3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampere equation. These results are proved using techniques from contact geometry.