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

2021/11/22 by Naff, Keaton, Weber, Brian
#34A12 #53B20 #53C25 #53C55 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.11150

Abstract

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 has a conformally Kähler (1,1) form.

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