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It is known that the almost-Kaehler anti-self-dual metrics on a given4-manifold sweep out an open subset in the moduli space of anti-self-dualmetrics. However, we show here by example that this subset is not generallyclosed, and so need not sweep out entire connected components in the modulispace. Our construction hinges on an unexpected link between harmonic functionson certain hyperbolic 3-manifolds and self-dual harmonic 2-forms on associated4-manifolds.

Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry