Einstein Spaces and Special Geometry

Einstein manifolds have played a central role in Riemannian geometry and mathematical physics over the last 100 years. Known examples of Einstein manifolds can be found within the classes of special geometry (for example Hermitian, algebraically special) or special holonomy metrics (for example hyperkähler, G2). Einstein manifolds appear naturally as possible final states and singularity models for the Ricci flow. In mathematical physics, Riemannian Einstein manifolds appear as gravitational instantons in quantum theories of gravity.

In spite of the classical results in the field, and much recent progress, many important questions about construction, classification, existence and uniqueness of Einstein manifolds remain wide open. For example, there is no known obstruction to the existence of Einstein metrics on compact manifolds of dimension n ≥ 5. A classical open question is whether all compact Ricci-flat manifold have special holonomy. There are also many open problems concerning the geometry of complete Ricci-flat with specific asymptotic geometries. For example, it is not known if all 4-dimensional Ricci-flat ALE manifolds are hyperkähler. Recent work has raised the question of whether there are any non-Hermitian gravitational instantons. Further important open problems concern the stability of Einstein manifolds, as well as concentration and bubbling phenomena at boundaries of moduli spaces of Einstein metrics.