On Chen-Teo geometries with cosmological constant

The Chen-Teo geometry is a Riemannian, Ricci-flat ALF 4-manifold, containing an AF gravitational instanton that gives the first counterexample to the Euclidean black hole uniqueness conjecture.

We investigate the problem of constructing an Einstein analogue with a non-zero cosmological constant $\lambda$.

We show that the solution is either the Plebański-Demiański metric with $\lambda$, or it has an anti-self-dual Weyl tensor.

We study the latter case in detail: we prove that for $\lambda<0$, there is a conformal infinity separating two asymptotically hyperbolic metrics; we show that one of them is globally conformal to an ALE scalar-flat Kähler metric; we construct gravitational instantons with different topologies; and we show that the geometry is a 4-pole solution in the Calderbank-Pedersen classification.