Selfdual Einstein metrics with torus symmetry

Abstract. It is well known that any 4-dimensional hyperkahler metric with two com-muting Killing elds may be obtained explicitly, via the Gibbons{Hawking Ansatz, from a harmonic function invariant under a Killing eld on R3. In this paper, we nd all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing elds. 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 ' hyper-bolic eigenfunctions yield explicit formulae for the quaternion-kahler quotients of HPm1 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-kahler quotients of HP2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unied framework, and give new complete examples. 1