Null cones from I+ and Legendre submanifolds

It is shown that the main variable Z of the null surface formulation of GR is the generating family of a constrained Lagrange submanifold that lives o n the energy surface H = 0 and that its level surfaces Z = const yield Lege ndre submanifolds on that energy surface. Thus, the singularity structure o f past null cones with apex at I+ is obtained by studying the projection ma p of the Legendre submanifolds to the configuration space. The behavior of the coordinate system defined by the variable Z at the caustic points is an alyzed. It is shown that a single function Z(xa, zeta, <(zeta)over bar>) ca nnot generate the conformal structure of an asymptotically flat space-time that satisfies the generic and weak energy condition. (C) 1999 American Ins titute of Physics. [S0022-2488(99)02105-2].