On a class of four-dimensional minimum energy metrics and hyperbolic geometry

A general problem in Differential Geometry is that how we can understand the topology of a 4-manifold in terms of 'optimal', 'canonical' or 'best' metrics. It is not easy to define what these are, but such 'best' metrics should be the ones easily perceivable to us, for instance it should have much symmetry or satisfies meaningful equations.;We consider minimum energy or critical metrics of two natural energy functionals in a natural moduli space of metrics; a scale invariant energy ;We start by discussing the Kahler critical metrics of two functionals. We then focus on 'self-dual' metrics which give minimum energies of the above functionals. We construct explicit hyperbolic ansatz self-dual metrics with semi-free conformal ;We observe that, from the explicitness of the metrics, we can read off the scalar curvature behavior without difficulty. Our construction includes all nonnegative scalar curved self-dual metrics with semi-free action. We also discuss a problem concerning scalar curvatures on the moduli space of self-dual conformal structures.;We also discuss examples in the context of minimum energy metrics of the above functionals.