Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay.
In this talk I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian.
This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.
Lars Andersson (Golm): Gravitational instantons and special geometry
