Curvature and Smooth Topology in Dimension Four

TL;DR

This paper explores the relationship between curvature estimates and smooth topology in four-dimensional manifolds, highlighting new bounds derived from Seiberg-Witten theory and their implications for Einstein and Kähler geometries.

Contribution

It presents a comprehensive overview of curvature estimates in four dimensions and investigates their optimality in relation to a conjecture in Kähler geometry.

Findings

Curvature estimates impose constraints on scalar curvature for manifolds with non-trivial Seiberg-Witten invariants.

These estimates extend to other curvature tensor components and influence Einstein manifold theory.

The paper discusses the potential optimality of these bounds in the context of Kähler geometry conjectures.

Abstract

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 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 Kaehler geometry.