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

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 CM...

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Published inSymmetry, integrability and geometry, methods and applications Vol. 9; p. 075
Main Author Sheftel, Mikhail B.
Format Journal Article
LanguageEnglish
Published Kiev National Academy of Sciences of Ukraine 27.11.2013
National Academy of Science of Ukraine
Subjects
anti-self-dual gravity
Boyer-Finley equation
group foliation
Monge-Ampère equation
non-invariant solutions
partner symmetries
Ricci-flat metric
symmetry reduction
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Summary: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. [ProQuest: [...] denotes formulae omitted.]
ISSN:1815-0659
1815-0659
DOI:10.3842/SIGMA.2013.075