Journal of Differential Geometry

Selfdual Einstein Metrics with Torus Symmetry

David M.J. Calderbank and Henrik Pedersen

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It is well-known that any 4-dimensional hyperkähler 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 • 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-kähler quotients of • Pm—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-kähler quotients of • P2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.

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J. Differential Geom., Volume 60, Number 3 (2002), 485-521.

First available in Project Euclid: 20 July 2004

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Calderbank, David M.J.; Pedersen, Henrik. Selfdual Einstein Metrics with Torus Symmetry. J. Differential Geom. 60 (2002), no. 3, 485--521. doi:10.4310/jdg/1090351125.

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