Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry
Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry
ÌýIf a smooth manifoldÌýMÌýadmits a symplectic form, it also admitsÌýRiemannian metrics g that are related to the symplectic form by means ofÌýanÌýadapted almost-complex structure. Such metrics are said to be almost-ÌýKaehler, because they are Kaehler if and only if the almost complex structureÌýisÌýintegrable. IfÌýMÌýis compact and 4-dimensional, one can then show that theÌýconformal classes of almost-Kaehler metrics sweep out an open subset in theÌýspace of the conformal classes. This provides a natural tool for exploringÌýdifficult global problems in 4-dimensional conformal geometry, leading toÌýnon-trivial results and motivating broader conjectures in the subject.
However, this technique certainly has its limitations. For example, if aÌý4-manifold admits scalar-flat Kaehler metrics, these can be deformed intoÌýanti-self-dual almost-Kaehler metrics, and these then sweep out an openÌýset in the moduli space of anti-self-dual conformal structures. One mightÌýsomehow hope thatÌýthis subset would also turn out to be closed, and soÌýsweep out entire connected components in the moduli space. Alas, however,Ìýthis simply isn’t true! In thisÌýtalk, I’ll explain recent joint work with ChrisÌýBishop that constructs a large hierarchy of counter-examples by studyingÌýthe limit sets of quasi-FuchsianÌýgroups.