Solutions and singularities of the semigeostrophic equations via the geometry of Lagrangian submanifolds

By using Monge-Ampère geometry, we study the singular structure of a class of nonlinear Monge-Ampère equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Ampère geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Ampère equation, while singularities and elliptic-hyperbolic transitions are revealed by degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.