The Curvature of 4-Manifolds

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Abstract Award: DMS-1205953 Principal Investigator: Claude R. LeBrun The principal investigator plans to study a family of related global problems in 4-dimensional Riemannian geometry. The primary goal of the proposed research is to discover and develop fundamental links between the curvature of a Riemannian 4-manifold and the differential topology of the underlying space. The main problems to be investigated focus on the existence, uniqueness, and detailed structure of several classes of canonical metrics. Topics of study will include Einstein metrics, extremal Kaehler metrics, Bach-flat metrics, and self-dual metrics. Some related problems in higher dimensions will also be considered, in anticipation of the discovery of further fundamental differences between the 4-dimensional and higher-dimensional settings. This research program aims both to explore fundamental issues in the mathematical field of differential geometry, and to discover new links between mathematics and theoretical physics. Much of the planned research activity takes its inspiration from current attempts to bridge the gulf separating Einstein's theory of gravitation from the quantum field theories that describe the forces of nature on a microscopic scale. A major focus of the research is the influence of the large-scale (topological) structure of a 4-dimensional universe on whether or not it admits Riemannian-signature solutions of Einstein's gravitational field equations.