Monge-Amp\`ere Systems with Lagrangian Pairs

TL;DR

This paper introduces classes of Monge-Ampère systems using Lagrangian contact structures, explores their geometric properties, and establishes symmetry bounds, providing a unified framework for various geometric equations.

Contribution

It defines decomposable and bi-decomposable Monge-Ampère systems within a geometric setting and analyzes their symmetry properties and classifications.

Findings

Lagrangian pairs uniquely determine bi-decomposable systems for ≥3 variables.

Several homogeneous Monge-Ampère systems naturally arise in geometry.

Sharp bounds on symmetry dimensions are established for these systems.

Abstract

The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables $\geq 3$. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Amp\`ere systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Amp\`ere systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).