Toric, \(U(2)\), and LeBrun metrics

  • Brian Weber Department of Mathematics, ShanghaiTech University, 319 Yueyang Road, Xuhui District, Shanghai, China.
Keywords: Differential geometry, Kähler geometry, canonical metrics, ansatz


The LeBrun ansatz was designed for scalar-flat Kähler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is Kähler, scalar-flat, or extremal Kähler. Second, toric Kähler metrics (such as the generalized Taub-NUTs) and \(U(2)\)-invariant metrics (such as the Fubini-Study or Page metrics) are certainly expressible in the LeBrun ansatz. We give general formulas for such transitions. We close the paper with examples, and find expressions for two examples — the exceptional half-plane metric and the Page metric — in terms of the LeBrun ansatz.


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How to Cite
B. Weber, “Toric, \(U(2)\), and LeBrun metrics”, CUBO, A Mathematical Journal, vol. 22, no. 3, pp. 395–410, Dec. 2020.