Selfdual Einstein metrics with torus symmetry
Abstract
It is well known that any 4dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the GibbonsHawking 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 EinsteinWeyl spaces found by Ward, and then show that certain `multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternionkahler quotients of HP(m1) by an (m2)torus first studied by Galicki and Lawson. As a consequence we are able to place the wellknown cohomogeneity one metrics, the quaternionkahler 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2001
 arXiv:
 arXiv:math/0105263
 Bibcode:
 2001math......5263C
 Keywords:

 Differential Geometry;
 Mathematical Physics;
 53C25;
 53C26;
 53A30
 EPrint:
 23 pages, 3 figures, AMSLaTeX and diagrams.sty