The Riemannian Geometry of 4-Manifolds

Einstein's equations originated in the general theory of relativity, where they govern the gravitational field by constraining the geometry of 4-dimensional space-time. The research funded by this grant will attempt to deepen our understanding of the "Riemannian" version of these equations, a context in which there is no difference between space and time directions. One central problem is to understand whether two solutions of these equations on a closed 4-dimensional space can always be continuously deformed into one another. Another key problem is to completely understand which 4-dimensional spaces carry solutions of these equations. Answers to these and related questions would be of fundamental interest not only in pure mathematics, but also in theoretical physics, where such problems arise in connection with theories of quantum gravity. The principal investigator will study a family of related problems in 4-dimensional global Riemannian geometry, focusing on existence, moduli, and detailed structure of Einstein metrics and other Bach-flat metrics on smooth compact 4-manifolds. The primary goal of this research program is to discover and elucidate fundamental links between the curvature of a Riemannian 4-manifold and the differential topology of the underlying space. The interplay between geometry and topology that occurs in dimension four is qualitatively different from what occurs other dimensions, and a deeper understanding of this phenomenon could have significant repercussions in differential topology. Related problems in higher-dimensional geometry will therefore also be considered when it becomes helpful to frame key issues in a broader context.