Some Rigidity Results Related to Monge-Ampere Functions

The space of Monge Ampere functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map u (sic) Det D(2)u is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge-Ampere functions. We also prove that if a Monge-Ampere function u on a bounded set Omega subset of R-2 satisfies the equation Det D(2)u = 0 in a particular weak sense, then the graph of u is a developable surface, and moreover u enjoys somewhat better regularity properties than an arbitrary Monge Ampere function of 2 variables.