Monge-Ampère systems with Lagrangian pairs

Goo Ishikawa, Yoshinori Machida

Published 2015-03-05Version 1

The classes of Monge-Amp{\`e}re systems, decomposable and bi-decomposable Monge-Amp{\`e}re 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{\`e}re systems with Lagrangian pairs arise naturally in various geometries. Moreover the best possible estimates are provided for the symmetries on decomposable and bi-decomposable Monge-Amp{\`e}re systems, in terms of the geometric structure.