  We prove certain weak or idealized existence results for minimizers of the
natural quadratic curvature functionals on the space of metrics on 4-manifolds.
Overall, we try to exhibit the relations with the picture in 3-dimensions
provided by Thurston geometrization. The existence results apply in particular
to the structure of the moduli spaces of such metrics, and as such generalize
most of the known results for moduli of Einstein metrics.
