If a smooth manifold M admits a symplectic form, it also admits Riemannian metrics g that are related to the symplectic form by means of an adapted almost-complex structure. Such metrics are said to be almost- Kaehler, because they are Kaehler if and only if the almost complex structure is integrable. If M is compact and 4-dimensional, one can then show that the conformal classes of almost-Kaehler metrics sweep out an open subset in the space of the conformal classes. This provides a natural tool for exploring difficult global problems in 4-dimensional conformal geometry, leading to non-trivial results and motivating broader conjectures in the subject.

However, this technique certainly has its limitations. For example, if a 4-manifold admits scalar-flat Kaehler metrics, these can be deformed into anti-self-dual almost-Kaehler metrics, and these then sweep out an open set in the moduli space of anti-self-dual conformal structures. One might somehow hope that this subset would also turn out to be closed, and so sweep out entire connected components in the moduli space. Alas, however, this simply isn’t true! In this talk, I’ll explain recent joint work with Chris Bishop that constructs a large hierarchy of counter-examples by studying the limit sets of quasi-Fuchsian groups.