Atlantis Press Paper Details: Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three- and twodimensional 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 may be possible to construct four-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors.

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title:		Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations
publication:		JNMP
volume-issue:		15 - supplement 3
pages:		385 - 395
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.2008.15.s3.37 (how to use a DOI)
author(s):		M. B. Sheftel, A. A. Malykh
publication date:		October 2008
abstract:		We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three- and twodimensional 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 may be possible to construct four-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors.
copyright:		© Atlantis Press. This article is distributed under the terms of the Creative Commons Attribution License, which permits non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
full text:		37_NEEDS_Sheftel.pdf (71 K)