Einstein metrics from the Calabi ansatz via Derdziński duality

Drawing on results of Derdziński’s from the 80’s, we classify conformally Kähler, $U(2)$-invariant, Einstein metrics on the total space of $\mathcal{O}(-m)$, for all $m\in \mathbb{N}$. This yields infinitely many $1$-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface ${\mathbb{H}}_m$ with a cone singularity along the “divisor at infinity”. This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci–flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}{\overline{\mathbb{C}\mathbb{P}}}^2$ when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric. Such metrics cannot be obtained by applying Derdziński’s theorem.