Solutions and Singularities of the Semigeostrophic Equations via the Geometry of Lagrangian Submanifolds

TL;DR

This paper explores the geometric structure of certain Monge-Ampère equations in 3D, revealing how metric signatures classify equations and how degeneracies indicate singularities and transitions, with applications to semigeostrophic fluid models.

Contribution

It extends Monge-Ampère geometry by linking metric signatures on Lagrangian submanifolds to equation classification and singularity analysis in geophysical fluid dynamics.

Findings

Metric signature classifies Monge-Ampère equations.

Degeneracies of the metric reveal singularities.

Application to semigeostrophic equations illustrates theory.

Abstract

Using Monge-Amp\`ere geometry, we study the singular structure of a class of nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere 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-Amp\`ere equation, while singularities and elliptic-hyperbolic transitions are revealed by the degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.