When is a solution to the degenerate Monge-Ampere equation developable?

I will talk about a few results in which the solutions to the Monge-Ampere equations in 2 or higher dimensions show rigid or flexible behavior depending on the regularity of the solution. As an example, the graph of a $W^{2,2}$ or a $C^{1,2/3+}$ weak solution to the degenerate Monge-Ampere equation in two dimensions is developable (rigid behavior), while, adapting the methods of Nash and Kuiper, one can show that developability fails for $C^{1,1/7-}$ weak solutions (flexible behavior). Moreover, it can be shown that the latter solutions are dense in the class of all continuous functions! Many problems remain open. The results are joint works with Bob Jerrard and Marta Lewicka.