Selfdual Einstein metrics with torus symmetry
Abstract
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.
Cite
@article{arxiv.math/0105263,
title = {Selfdual Einstein metrics with torus symmetry},
author = {David M. J. Calderbank and Henrik Pedersen},
journal= {arXiv preprint arXiv:math/0105263},
year = {2007}
}
Comments
23 pages, 3 figures, AMS-LaTeX and diagrams.sty