Toric, U(2), and LeBrun metrics

Abstract

The LeBrun ansatz was designed for scalar-flat Kahler 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 Kahler, scalar-flat, or extremal Kahler. Second, toric Kahler 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.