The moment map on the space of symplectic 3D MongeAmpère equations
Abstract
For any secondorder scalar PDE $\mathcal{E}$ 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 $\mathcal{E}$, a subbundle of the contact distribution of the underlying contact manifold $J^1$, consisting of conic varieties. We call it the contact cone structure associated with $\mathcal{E}$. We then focus on symplectic MongeAmpère equations in 3 independent variables, that are naturally parametrized by a 13dimensional real projective space. If we pass to the field of complex numbers $\mathbb{C}$, this projective space turns out to be the projectivization of the 14dimensional irreducible representation of the simple Lie group $\mathsf{Sp}(6,\mathbb{C})$: the associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D MongeAmpère 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 $\varpi$, herewith called the cocharacteristic variety, and the contact cone structure of a 3D MongeAmpère equation $\mathcal{E}$: under the hypothesis of nondegenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually nonequivalent quadratic forms on a $6$dimensional symplectic space, which has an interest on its own.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 DOI:
 10.48550/arXiv.2105.06675
 arXiv:
 arXiv:2105.06675
 Bibcode:
 2021arXiv210506675G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Analysis of PDEs;
 Mathematics  Representation Theory;
 Mathematics  Symplectic Geometry;
 35A30;
 58A20;
 58J70
 EPrint:
 Accepted by Advances in Differential Equations