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

For \(U(2)\)-invariant 4-metrics, we show that the \(B^t\)-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.