On the integrability of symplectic MongeAmpère equations
Abstract
Let u be a function of n independent variables x^{1},…,x^{n}, and let U=(u) be the Hessian matrix of u. The symplectic MongeAmpère equation is defined as a linear relation among all possible minors of U. Particular examples include the equation detU=1 governing improper affine spheres and the socalled heavenly equation, u_{13}u_{24}u_{23}u_{14}=1, describing selfdual Ricciflat 4manifolds. In this paper we classify integrable symplectic MongeAmpère equations in four dimensions (for n=3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of 'maximally singular' hyperplane sections of the Plücker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F(u)=0 in more than three dimensions is necessarily of the symplectic MongeAmpère type.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 October 2010
 DOI:
 10.1016/j.geomphys.2010.05.009
 arXiv:
 arXiv:0910.3407
 Bibcode:
 2010JGP....60.1604D
 Keywords:

 Mathematics  Differential Geometry;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 35L70;
 35Q58;
 35Q75;
 53A20;
 53D99;
 53Z05
 EPrint:
 20 pages