The Seiberg-Witten invariants of manifolds with wells of negative   curvature
  We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in the paper.) The idea is to combine the method of `semigroup domination' with the Weitzenbock formula. The same curvature hypothesis implies the vanishing of the Seiberg-Witten invariant of any finite covering space. We also show that, in high dimensions, a spin manifold with mostly positive scalar curvature in fact admits a metric of positive scalar curvature. 
Daniel Ruberman