Complexified Hamiltonian symplectomorphisms and solutions of the homegeneous complex Monge-Ampere equation

The geodesics for the Mabuchi metric on the space of Kaehler metrics on a manifold correspond to solutions of the homogeneous complex Monge-Ampere (HCMA) equation. We will describe a method for reducing the Cauchy problem for the HCMA equation with analytic initial data to finding a related Hamiltonian flow followed by a "complexification". Examples and applications will be discussed.  Work in collaboration with J.P. Nunes and T. Reis