From the same journal

From the same journal

Selfdual Einstein metrics with torus symmetry

Research output: Contribution to journalArticlepeer-review


  • H. Pedersen
  • D.M.J. Calderbank


Publication details

JournalJournal of Differential Geometry
DatePublished - Mar 2002
Issue number3
Number of pages36
Pages (from-to)485-521
Original languageEnglish


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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