Seminarium Teorii Względności i Grawitacji

A gravitational instanton is a four-dimensional, complete, Ricci-flat, Riemannian manifold with prescribed curvature decay at infinity. This includes the notable example of the euclidean Kerr instanton, which was conjectured by Gibbons, Hawking and Lapedes to be the unique asymptotically flat solution in this class. Remarkably, over 30 years later, Chen and Teo constructed an explicit counterexample to this Riemannian version of the no-hair conjecture. Interestingly, both of these examples are toric and possess a Hermitian (non-Kahler) structure. I will discuss recent progress on the classification of generic toric instantons that are ALF (asymptotically locally flat) and ALE (asymptotically locally euclidean), including the special subclass that are Hermitian for which a complete classification is possible.