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

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-Ampère 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-Kähler 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.

Keywords

• Monge-Ampère equation
• Boyer-Finley equation
• partner symmetries
• symmetry reduction
• non-invariant solutions
• group foliation
• anti-self-dual gravity
• Ricci-flat metric