Exotic Smooth Structures on Four-Manifolds with Non-Positive Sectional Curvature via Seiberg-Witten Theory

This monograph investigates the interplay between non-positive sectional curvature and the existence of exotic smooth structures on compact four-manifolds. Utilizing the Seiberg-Witten invariants, we analyze the obstruction mechanisms for smooth structures on topological manifolds that admit locally symmetric metrics of non-compact type. We specifically examine the case of complex hyperbolic surfaces and their exotic counterparts constructed via Fintushel-Stern knot surgery. By invoking the Weitzenböck formulae for the Dirac operator and coupling them with LeBrun's curvature estimates, we derive rigorous bounds on the Seiberg-Witten basic classes for manifolds satisfying strict curvature inequalities. Furthermore, we discuss the rigidity of the standard smooth structure on quotients of the complex hyperbolic plane under the assumption of Einstein metrics, contrasting this with the flexibility exhibited in the absence of curvature constraints. The paper culminates in a discussion of the non-existence of metrics with non-positive sectional curvature on certain families of exotic manifolds with non-trivial Seiberg-Witten invariants, thereby establishing a link between smooth topology and Riemannian geometry.