Stability under deformations of extremal almost-Kähler metrics in dimension $4.$

Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact $4$-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler 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.

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Published 1 January 2010