Aspects of the Seiberg-Witten Equations on Manifolds with Cusps

Abstract

In this work we study several geometrical and analytical aspects arising from the study of the Seiberg-Witten equations on manifolds with cusps. We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riemannian metrics of nonpositive curvature, but which do not admit K\ahler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients. The proof of these results use a Riemannian cusps closing technique developed by Hummel and Schroeder. Using a construction due to Biquard, we derive an obstruction to the existence of cuspidal Einstein metrics on finite-volume complex surfaces. This generalizes a theorem of LeBrun for compact complex surfaces. As in the compact case, such a result relies on a Seiberg-Witten scalar curvature estimate. Then, the obstruction is made explicit on some examples. Finally, we study the Seiberg-Witten equations on noncompact manifolds which are diffeomorphic to the product of two hyperbolic Riemann surfaces. By extending some constructions of Biquard and Rollin, we show how to construct irreducible solutions of the Seiberg-Witten equations for \emph{any} metric of finite volume which has a ``nice'' behavior at infinity. We conclude by giving the finite volume generalization of some celebrated results of LeBrun.