On the integrability of symplectic Monge-Ampère equations. (English) Zbl 1195.35109
Summary: Let \(u\) be a function of \(n\) independent variables \(x^{1},\ldots ,x^n\), and let \(U=(u_{ij})\) be the Hessian matrix of \(u\). The symplectic Monge-Ampère equation is defined as a linear relation among all possible minors of \(U\). Particular examples include the equation \(\det U=1\) governing improper affine spheres and the so-called heavenly equation, \(u_{13}u_{24} - u_{23}u_{14}=1\), describing self-dual Ricci-flat 4-manifolds. In this paper, we classify integrable symplectic Monge-Ampère 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 \(F(u_{ij})=0\) in more than three dimensions is necessarily of symplectic Monge-Ampère type.
MSC:
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35Q75 | PDEs in connection with relativity and gravitational theory |
| 35J96 | Monge-Ampère equations |
| 35K96 | Parabolic Monge-Ampère equations |
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