STABILITY UNDER DEFORMATIONS OF EXTREMAL ALMOST-KAHLER METRICS IN DIMENSION 4.

Given a path of almost-Kahler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kahler metric is an extremal Kahler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.