Lars Andersson: Gravitational instantons and special geometry

Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. Classical examples include the Taub-NUT and the Euclidean Kerr instanton. A classification of half-flat instantons is known but the uniqueness problem remains open in general. In this talk I will present some recent results the classification of $S^1$-symmetric instantons obtained using an identity of Israel-Robinson type and the $G$-signature theorem, together with recent results on instantons with special geometry.