Covariant symplectic structure of the complex Monge-Ampère equation

Abstract

The complex Monge-Ampère equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of 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. However, we point out that Hilbert's Lagrangian is a divergence for Kähler metrics. Nevertheless, using the surface terms in the Hilbert Lagrangian we obtain the second Witten-Zuckerman symplectic 2-form for complex dimension>2.

Physics Letters A 268 (2000) 293

References in corpus (2)

• Gravitational instantons admit hyper-Kähler structure
• Hamiltonian structure of real Monge-Ampère equations

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