Toric, U(2), and LeBrun metrics

Abstract

The LeBrun ansatz was designed for scalar-flat K\"ahler 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 K\"ahler, scalar-flat, or extremal K\"ahler. Second, toric K\"ahler 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.

Keywords

• differential geometry
• kähler geometry
• canonical metrics
• ansatz