The moment map on the space of symplectic 3D Monge-Amp\`ere equations

TL;DR

This paper explores the geometric structure of symplectic Monge-Ampère equations in three variables using moment maps, revealing a deep link between their contact cone structures and cocharacteristic varieties through Lie group representations.

Contribution

It introduces a moment map framework for 3D symplectic Monge-Ampère equations, connecting contact cone structures with cocharacteristic varieties via Lie group theory.

Findings

The moment map maps symplectic Monge-Ampère equations to quadratic forms.

The contact cone structure coincides with the cocharacteristic variety under certain conditions.

A complete classification of quadratic forms on symplectic spaces is provided.

Abstract

For any second-order scalar PDE $\mathcal{E}$ in one unknown function, that we interpret as a hypersurface of a second-order jet space $J^2$, we construct, by means of the characteristics of $\mathcal{E}$, a sub-bundle of the contact distribution of the underlying contact manifold $J^1$, consisting of conic varieties. We call it the contact cone structure associated with $\mathcal{E}$. We then focus on symplectic Monge-Amp\`ere equations in 3 independent variables, that are naturally parametrized by a 13-dimensional real projective space. If we pass to the field of complex numbers $\mathbb{C}$, this projective space turns out to be the projectivization of the 14-dimensional irreducible representation of the simple Lie group $\mathsf{Sp}(6,\mathbb{C})$: the associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D Monge-Amp\`ere equations to the projectivization of the space of quadratic forms on a $6$-dimensional symplectic vector space. We study in details the relationship between the zero locus of the image of $\varpi$, herewith called the cocharacteristic variety, and the contact cone structure of a 3D Monge-Amp\`ere equation $\mathcal{E}$: under the hypothesis of non-degenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually non-equivalent quadratic forms on a $6$-dimensional symplectic space, which has an interest on its own.