Families of degenerating Poincaré–Einstein metrics on \(\mathbb {R}^4\)

Abstract

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.