The Complex and Conformal Geometry of 4-Manifolds

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

In the general theory of relativity, Einstein's equations govern the gravitational field by constraining the geometry of a 4-dimensional space-time. These equations are the prototype for an important family of geometric partial differential equations that will be investigated under this grant, in the "Riemannian" setting where there is no difference between space and time. A principal objective of the funded research will be to determine precisely which 4-dimensional spaces carry solutions of these equations. Progress in this area could have significant repercussions outside of pure mathematics, especially because many of the geometrical questions to be investigated are intrinsically linked to problems in classical and quantum gravity, as well to other areas of theoretical physics. Graduate students will be trained through participation in the research activities. More specifically, this project focuses on a family of related problems in 4-dimensional global Riemannian geometry, using techniques of complex, conformal, Riemannian, Kaehler, and symplectic geometry. Topics to be investigated include Einstein metrics, the Einstein-Maxwell equations, ALE manifolds, scalar-flat Kaehler metrics, Bach-flat 4-manifolds, Poincare-Einstein metrics, and various twistor constructions. The main goal is to discover and elucidate fundamental links between curvature and topology, especially in dimension four. One focus is the construction of canonical metrics on compact manifolds, primarily by means of Kaehler and conformal methods. Another focus is on understanding the relevant moduli spaces of solutions, along with associated bubbling phenomena. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.