Scalar--flat K\"ahler metrics with conformal Bianchi V symmetry

TL;DR

This paper constructs all scalar-flat Kähler metrics in four dimensions that are invariant under conformal Bianchi V symmetry, using twistor theory and isomonodromic problems, and characterizes solutions via symmetry analysis.

Contribution

It provides a complete classification of scalar-flat Kähler metrics with conformal Bianchi V symmetry, explicitly expressed through Bessel functions, and links them to the SU(∞) Toda equation.

Findings

Explicit metrics in terms of Bessel functions

Connection with the LeBrun ansatz and Toda equation symmetries

Complete classification of conformal Bianchi V scalar-flat Kähler metrics

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.