Ricci curvature, minimal volumes, and Seiberg-Witten theory

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with non-trivial Seiberg-Witten invariants. These allow one, for example, to exactly comput e the infimum of the L-2-norm of Ricci curvature for any complex surface of general type. We are also able to show that the standard metric on any com plex-hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of th e scalar curvature. These estimates also imply new non-existence results fo r Einstein metrics.