The moment map on the space of symplectic 3D Monge-Ampère equations

Abstract

For any 2 nd order scalar PDE E in one unknown function, that we interpret as a hypersurface of a secondorder jet space J 2 , we construct, by means of the characteristics of E, a sub-bundle of the contact distribution of the underlying contact manifold J 1 , consisting of conic varieties. We call it the contact cone structure associated with E. We then focus on symplectic Monge-Ampère equations in 3 independent variables, that are naturally parametrized by a 13-dimensional real projective space. If we pass to the field of complex numbers C, this projective space turns out to be the projectivization of the 14-dimensional irreducible representation of the simple Lie group Sp(6, C): the associated moment map allows to define a rational map from the space of symplectic 3D Monge-Ampère equations to the projectivization of the space of quadratic forms on a 6-dimensional symplectic vector space. We study in details the relationship between the zero locus of the image of , herewith called the cocharacteristic variety, and the contact cone structure of a 3D Monge-Ampère equation E: under the hypothesis of non-degenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually non-equivalent quadratic forms on a 6-dimensional symplectic space, which has an interest on its own. Contents 1. Introduction 1 2. Preliminaries 3 3. The contact cone structure associated with a second-order PDE 7 4. Quadric contact cone structures associated with 3D Monge-Ampère equations 10 5. Reconstructing a second-order PDE from a contact cone structure 13 6. The space PΛ 3 0 (C) of symplectic 3D Monge-Ampère equations 15 7. Normal forms of quadratic forms on C with respect to Sp(C) 18 8. The moment map on the space of Monge-Ampère equations 25 9. Hyperplane sections of the Lagrangian Grassmannian LGr(3, C) 30 10. Conclusions and perspectives 35 References 35