Canonical metrics and ambiKähler structures on 4-manifolds with \(U(2)\) symmetry

For \ (U (2) \) -invariant 4-metrics, we show that the \ (Bᵗ\) -flat metrics are very different from the other canonical metrics (Bach-flat, Einstein, extremal Kähler, etc. ) We show every \ (U (2) \) -invariant metric is conformal to two separate Kähler metrics, leading to ambiKähler structures. Using this observation we find new complete extremal Kähler metrics on the total spaces of \ (O (-1) \) and \ (O (+1) \) that are conformal to the Taub-bolt metric. In addition to its usual hyperKähler structure, the Taub-NUT's conformal class contains two additional complete Kähler metrics that make up an ambi-Kähler pair, making five independent compatible complex structures for the Taub-NUT, each of which is conformally Kähler.