Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

TL;DR

This paper introduces advanced symmetry methods to derive noninvariant solutions of the complex Monge-Ampère equation, leading to new Ricci-flat anti-self-dual Einstein-Kähler metrics without Killing vectors, free of singularities in certain domains.

Contribution

It develops a novel combination of partner symmetries and group foliation techniques to construct noninvariant solutions and associated metrics without Killing vectors.

Findings

Derived a new noninvariant solution of CMA.

Constructed Ricci-flat anti-self-dual Einstein-Kähler metrics without Killing vectors.

Found metrics free of singularities in specific domains.

Abstract

We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Amp\`ere equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-K\"ahler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.