Extremal almost-Kahler metrics and Seiberg-Witten theory

We show that there exist compact non-Kahler almost-Kahler 4-manifolds whose metrics minimize L-2-norm of (2/3) s + 2w among all metrics compatible with a fixed decomposition H-2(M, bb R = H+ + H-, where s is the scalar curvature and w is the lowest eigenvalue of self-dual Weyl curvature at each point. In particular, the moduli space of such metrics modulo diffeomorphisms is infinite dimensional. This example also shows that LeBrun's estimate of L-2-norm of (1 - delta)s + delta . 6w on a compact oriented Riemannian 4-manifold with a nontrivial Seiberg-Witten invariant cannot be extended over delta = 1/3.