SYMPLECTIC STRUCTURES AND SYMMETRIES OF SOLUTIONS OF THE COMPLEX MONGE-AMPERE EQUATION

The graphs that arise from the gradients of solutions u of the homogen eous complex Monge-Ampere equation are characterized in terms of the n atural symplectic structure on the cotangent bundle. This characteriza tion is invariant under symplectic biholomorphisms. Using the symplect ic structures we construct symmetries (to be called Lempert transforma tions) for real valued functions u. which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampere equation and to transform the Poincare-Lelong equation and the partial derivative-equation. An exam ple of Lempert transform is given and the main theorem is applied to p rove regularity results for exterior nonlinear Dirichlet problem for t he homogeneous complex Monge-Ampere equation.