### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 081, 32 pages      arXiv:1503.01529      https://doi.org/10.3842/SIGMA.2015.081

### Monge-Ampère Systems with Lagrangian Pairs

Goo Ishikawa a and Yoshinori Machida b
a) Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
b) Numazu College of Technology, 3600 Ooka, Numazu-shi, Shizuoka, 410-8501, Japan

Received April 10, 2015, in final form October 05, 2015; Published online October 10, 2015

Abstract
The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère 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-Ampère 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-Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).

Key words: Hessian Monge-Ampère equation; non-degenerate three form; bi-Legendrian fibration; Lagrangian contact structure; geometric structure; simple graded Lie algebra.

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