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.

References in corpus (2)

• Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics
• Enhanced Euclidean supersymmetry, 11D supergravity and $SU(\infty )$ Toda equation

Cited by in corpus (2)

• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• Nonlocal symmetry of CMA generates ASD Ricci-flat metric with no Killing vectors