The Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds

Irreducible solutions to the Seiberg-Witten equations give a priori estimates for the total scalar curvature of the underlying 4-manifold. This was used by LeBrun in the late 90s to construct the first examples of closed 4-manifolds that satisfy the strict Hitchin-Thorpe inequality yet do not admit any Einstein metrics. In this talk, I will describe an extension of this story to the finite volume setting, where we consider complete metrics with asymptotically hyperbolic cusps. As an application we will construct an infinite family of finite volume 4-manifolds with $T^3$ ends that do not admit any asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei.