Covariant symplectic structure of the complex Monge–Ampère equation

Abstract

The complex Monge–Ampère equation is invariant under arbitrary holomorphic changes of the independent variables with unit Jacobian. We present its variational formulation where the action remains invariant under this infinite group. The new Lagrangian enables us to obtain the first symplectic 2-form for the complex Monge–Ampère equation in the framework of the covariant Witten–Zuckerman approach to symplectic structure. We base our considerations on a reformulation of the Witten–Zuckerman theory in terms of holomorphic differential forms. The first closed and conserved Witten–Zuckerman symplectic 2-form for the complex Monge–Ampère equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous.The connection of the complex Monge–Ampère equation with Ricci-flat Kähler geometry suggests the use of the Hilbert action principle as an alternative variational formulation. However, we point out that Hilbert's Lagrangian is a divergence for Kähler metrics and serves as a topological invariant rather than yielding the Euclideanized Einstein field equations. Nevertheless, since the Witten–Zuckerman theory employs only the boundary terms in the first variation of the action, Hilbert's Lagrangian can be used to obtain the second Witten–Zuckerman symplectic 2-form. This symplectic 2-form vanishes on shell, thus defining a Lagrangian submanifold. In its derivation the connection of the second symplectic 2-form with the complex Monge–Ampère equation is indirect but we show that it satisfies all the properties required of a symplectic 2-form for the complex elliptic, or hyperbolic Monge–Ampère equation when the dimension of the complex manifold is 3 or higher.The complex Monge–Ampère equation admits covariant bisymplectic structure for complex dimension 3, or higher. However, in the physically interesting case of n=2 we have only one symplectic 2-form.The extension of these results to the case of complex Monge–Ampère–Liouville equation is also presented.