On symplectic classification of effective 3-forms and Monge-Ampere equations

TL;DR

This paper classifies effective 3-forms under symplectomorphisms in six-dimensional space, identifies the special Lagrangian form within this classification, and provides a local classification for related Monge-Ampère equations.

Contribution

It completes the classification of effective 3-forms with constant coefficients under symplectic transformations and links these forms to Monge-Ampère equations.

Findings

Identified new normal forms for effective 3-forms.

Included the special Lagrangian form in the classification.

Established a local classification theorem for Monge-Ampère equations.

Abstract

We complete the list of normal forms for effective 3-forms with constant coefficients with respect to the natural action of symplectomorphisms in \mathbb{R}^6. We show that the 3-form which corresponds to the Special Lagrangian equation is among the new members of the classification. The symplectic symmetry algebras and their Cartan prolongations for these forms are computed and a local classification theorem for the corresponding Monge-Ampere equations is proved.