Solutions of the sDiff(2)Toda equation with SU(2) Symmetry

TL;DR

This paper derives the general solution to the sDiff(2)Toda equation with SU(2) symmetry, linking it to self-dual Einstein metrics and extending previous work on related geometric structures.

Contribution

It provides the first general solution to the sDiff(2)Toda equation with SU(2) symmetry, connecting it to self-dual Einstein metrics and generalizing earlier approaches.

Findings

Derived the general solution for the sDiff(2)Toda equation with SU(2) symmetry.

Connected solutions to self-dual Einstein metrics with Bianchi IX symmetry.

Extended previous work on Atiyah-Hitchin and Bianchi IX metrics.

Abstract

We present the general solution to the Plebanski equation for an H-space that admits Killing vectors for an entire SU(2) of symmetries, which is therefore also the general solution of the sDiff(2)Toda equation that allows these symmetries. Desiring these solutions as a bridge toward the future for yet more general solutions of the sDiff(2)Toda equation, we generalize the earlier work of Olivier, on the Atiyah-Hitchin metric, and re-formulate work of Babich and Korotkin, and Tod, on the Bianchi IX approach to a metric with an SU(2) of symmetries. We also give careful delineations of the conformal transformations required to ensure that a metric of Bianchi IX type has zero Ricci tensor, so that it is a self-dual, vacuum solution of the complex-valued version of Einstein's equations, as appropriate for the original Plebanski equation.