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Journal of Differential Geometry 60 (2002), 485-521.

Selfdual Einstein metrics with torus symmetry

David M.J. Calderbank & Henrik Pedersen

Abstract:

It is well-known that any $4$-dimensional hyper\kahl/ metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on $\R^3$. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain `multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-\kahl/ quotients of $\HP{m-1}$ by an $(m-2)$-torus studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-\kahl/ quotients of $\HP{2}$ (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.