Curvature and Smooth Topology in Dimension Four

Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question has a non-trivial Seiberg-Witten invariant. However, it has recently been discovered [20, 22] that similar statements also apply to other parts of the curvature tensor. This article presents the most salient aspects of these curvature estimates in a self-contained manner, and shows how they can be applied to the theory of Einstein manifolds. We then probe the issue of whether the known estimates are optimal by relating this question to a certain conjecture in Kähler geometry. ∗Supported in part by NSF grant DMS-9802722.