2025 Theses Doctoral

The Seiberg—Witten Equations and Asymptotically Hyperbolic Einstein Metrics

Xu, Alex; Lin, Francesco

In this thesis we study the Seiberg--Witten equations and Einstein metrics on finite volume noncompact 4-manifolds with asymptotically hyperbolic cusps. The problem of obstructing the existence of Einstein metrics for closed and oriented 4-manifolds with nonzero Seiberg--Witten invariant was pioneered by LeBrun. Various extensions of this problem in the noncompact setting were subsequently studied by Biquard, di Cerbo, and Rollin in the asymptotically complex hyperbolic case.

This dissertation extends this story in the asymptotically real hyperbolic setting where the ends of the manifolds are diffeomorphic to 𝑇³ x [0,∞). The main result of this dissertation is the construction of the first examples of noncompact 4-manifolds that does not admit any asymptotically hyperbolic Einstein metrics. Along the way, we also extend these techniques to the setting of the Pin ̄ (2) monopole equations developed by Nakamura.