Families of degenerating Poincaré-Einstein metrics on ${\mathbb{R}}^4$

We provide the first example of continuous families of Poincaré-Einstein metrics developing cusps on the trivial topology ${\mathbb{R}}^4$. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Plebański-Demiański. We additionally indicate how to construct similar examples on more complicated topologies.