Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

DOI

10.2991/jnmp.2008.15.s3.37How to use a DOI?

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

© 2008, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - M.B. Sheftel
AU  - A.A. Malykh
PY  - 2008
DA  - 2008/10/01
TI  - Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 385
EP  - 395
VL  - 15
IS  - supplement 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2008.15.s3.37
DO  - 10.2991/jnmp.2008.15.s3.37
ID  - Sheftel2008
ER  -