On the integrability of symplectic Monge-Amp\'ere equations

TL;DR

This paper classifies integrable symplectic Monge-Ampere equations in four dimensions, revealing their geometric structure and proposing a conjecture about their uniqueness in higher dimensions.

Contribution

It provides a classification of integrable symplectic Monge-Ampere equations in four dimensions and formulates a conjecture on their exclusivity in higher dimensions.

Findings

Classification of integrable symplectic Monge-Ampere equations in 4D

Geometric reformulation via hyperplane sections of Lagrangian Grassmannian

Conjecture that all higher-dimensional integrable equations are of this type

Abstract

Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the equation det U=1 governing improper affine spheres and the so-called heavenly equation, u_{13}u_{24}-u_{23}u_{14}=1, describing self-dual Ricci-flat 4-manifolds. In this paper we classify integrable symplectic Monge-Ampere equations in four dimensions (for n=3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of 'maximally singular' hyperplane sections of the Plucker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F(u_{ij})=0 in more than three dimensions is necessarily of the symplectic Monge-Ampere type.