We study the Seiberg-Witten equations on noncompact manifolds diffeomorphic to the product of two hyperbolic Riemann surfaces. First, we show how to construct irreducible solutions of the Seiberg-Witten equations for any metric of finite volume which has a "nice'' behavior at infinity. Then we compute the infimum of the L²-norm of scalar curvature on these spaces and give nonexistence results for Einstein metrics on blow-ups. This generalizes to the finite volume setting some well-known results of LeBrun.

This work has been partially supported by the Simons Foundation.