Einstein metrics from the Calabi ansatz via Derdzi\'nski duality

TL;DR

This paper classifies a broad family of conformally Kähler, $U(2)$-invariant Einstein metrics on line bundles, revealing new transitions between different geometric behaviors including hyperbolic, ALF, and singular metrics.

Contribution

It extends Derdziński's classification to new settings, constructing diverse Einstein metrics with various asymptotic behaviors and singularities, and explores geometric transitions between them.

Findings

Constructed infinitely many 1-parameter families of Einstein metrics.

Demonstrated the limit of cone angle Einstein metrics yields Ricci-flat ALF metrics.

Produced Einstein metrics conformal to scalar-flat Kähler metrics, not obtainable via Derdziński's theorem.

Abstract

Drawing on results of Derdzi\'nski's from the 80's, we classify conformally K\"ahler, $U(2)$-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 Poincar\'e 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 $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat K\"ahler metric. Such metrics cannot be obtained by applying Derdzi\'nski's theorem.