  We derive new, sharp lower bounds for certain curvature functionals on the
space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial
Seiberg-Witten invariant. These allow one, for example, to exactly compute the
infimum of the L2-norm of Ricci curvature for all complex surfaces of general
type. We are also able to show that the standard metric on any complex
hyperbolic 4-manifold minimizes volume among all metrics satisfying a
point-wise lower bound on sectional curvature plus suitable multiples of the
scalar curvature. These estimates also imply new non-existence results for
Einstein metrics.
