Seiberg-Witten equations on surfaces of logarithmic general type

Abstract

We study the Seiberg-Witten equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the Seiberg-Witten equations for any metric which is "asymptotic" to a Poincar\'e type metric at infinity. Then we compute a lower bound for the $L^{2}$-norm of scalar curvature on these spaces and give non-existence results for Einstein metrics on blow-ups.