Selfdual Einstein metrics with torus symmetry

Details Selfdual Einstein metrics with torus symmetry Selfdual Einstein metrics with torus symmetry

2001-05-31

arxiv.org/abs/math/0105263v1

J. Diff. Geom. 60 (2002) 485-521.

Mathematics

Differential Geometry

23 pages, 3 figures, AMS-LaTeX and diagrams.sty

Scientific paper

It is well known that any 4-dimensional hyperkahler 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-kahler quotients of HP(m-1) by an (m-2)-torus first studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kahler 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.

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