Scalar--flat Kähler metrics with conformal Bianchi V symmetry
arXiv:1010.2963 · doi:10.1088/0264-9381/28/12/125004
Abstract
We provide an affirmative answer to a question posed by Tod \cite{Tod:1995b}, and construct all four-dimensional Kahler metrics with vanishing scalar curvature which are invariant under the conformal action of Bianchi V group. The construction is based on the combination of twistor theory and the isomonodromic problem with two double poles. The resulting metrics are non-diagonal in the left-invariant basis and are explicitly given in terms of Bessel functions and their integrals. We also make a connection with the LeBrun ansatz, and characterise the associated solutions of the SU(\infty) Toda equation by the existence a non-abelian two-dimensional group of point symmetries.
Dedicated to Maciej Przanowski on the occasion of his 65th birthday. Minor corrections. To appear in CQG