Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates

TL;DR

This paper demonstrates that eigenfunctions serve as natural coordinates for self-dual Einstein spaces, revealing the general heavenly equation as the core governing equation and linking various known heavenly equations through a unified formalism.

Contribution

It introduces a formalism connecting eigenfunctions to the general heavenly equation and related heavenly equations, including new insights into their geometric and integrable structures.

Findings

Eigenfunctions form privileged coordinates for self-dual Einstein spaces.

The general heavenly equation is identified as the fundamental governing equation.

Connections between different heavenly equations and integrable systems are established.

Abstract

Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebanski's first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4+4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein-Weyl geometries.