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March, 2002 Selfdual Einstein Metrics with Torus Symmetry
David M.J. Calderbank, Henrik Pedersen
J. Differential Geom. 60(3): 485-521 (March, 2002). DOI: 10.4310/jdg/1090351125

Abstract

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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David M.J. Calderbank. Henrik Pedersen. "Selfdual Einstein Metrics with Torus Symmetry." J. Differential Geom. 60 (3) 485 - 521, March, 2002. https://doi.org/10.4310/jdg/1090351125

Information

Published: March, 2002
First available in Project Euclid: 20 July 2004

zbMATH: 1067.53034
MathSciNet: MR1950174
Digital Object Identifier: 10.4310/jdg/1090351125

Rights: Copyright © 2002 Lehigh University

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Vol.60 • No. 3 • March, 2002
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