Lift of Invariant to Non-Invariant Solutions of Complex Monge-Amp\`ere Equations

TL;DR

This paper demonstrates how partner symmetries can lift non-invariant solutions from lower-dimensional equations to the complex Monge-Ampère equations, enabling the construction of non-symmetric Ricci-flat metrics in four dimensions.

Contribution

It introduces a method using partner symmetries to generate non-invariant solutions of complex Monge-Ampère equations from lower-dimensional equations.

Findings

Lifted non-invariant solutions of CMA and HCMA from lower-dimensional equations.

Constructed four-dimensional Ricci-flat metrics with no Killing vectors.

Generated metrics with non-zero curvature tensors.

Abstract

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Amp\`ere equations (CMA and HCMA) provide a lift of non-invariant solutions of three- and two-dimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the two-dimensional Helmholtz equation to yield non-invariant solutions of CMA, and to non-invariant solutions of three-dimensional wave equation and three-dimensional hyperbolic Boyer-Finley equation to yield non-invariant solutions of HCMA. By using these solutions as metric potentials, it is possible to construct four-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors.