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Einstein–Maxwell Equations on Four-dimensional Lie Algebras

Published online by Cambridge University Press:  09 May 2019

Caner Koca
Affiliation:
Department of Mathematics, NYC College of Technology of CUNY, Brooklyn, NY 11021, USA Email: ckoca@citytech.cuny.edu
Mehdi Lejmi
Affiliation:
Department of Mathematics, Bronx Community College of CUNY, Bronx, NY 10453, USA Email: mehdi.lejmi@bcc.cuny.edu

Abstract

We classify up to automorphisms all left-invariant non-Einstein solutions to the Einstein–Maxwell equations on four-dimensional Lie algebras.

Type
Article
Copyright
© Canadian Mathematical Society 2019 

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Footnotes

The authors were supported in part by a PSC-CUNY research award #61768-00 49.

References

Apostolov, V., Calderbank, D. M. J., and Gauduchon, P., Ambitoric geometry I: Einstein metrics and extremal ambikähler structures . J. Reine Angew. Math. 721(2016), 109147. https://doi.org/10.1515/crelle-2014-0060 Google Scholar
Apostolov, V. and Maschler, G., Conformally Kähler, Einstein–Maxwell geometry . J. Eur. Math. Soc. 21(2019), no. 5, 13191360. https://doi.org/10.4171/JEMS/862 Google Scholar
Drăghici, T., On some 4-dimensional almost Kähler manifolds . Kodai Math. J. 18(1995), no. 1, 156168. https://doi.org/10.2996/kmj/1138043359 Google Scholar
Fino, A., Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor . Differential Geom. Appl. 23(2005), 1, 2637. https://doi.org/10.1016/j.difgeo.2005.03.003 Google Scholar
Futaki, A. and Ono, H., Conformally Einstein–Maxwell Kähler metrics and structure of the automorphism group . Math. Z. 292(2019), no. 1–2, 571589. https://doi.org/10.1007/s00209-018-2112-3 Google Scholar
Futaki, A. and Ono, H., On the existence problem of Einstein–Maxwell Kähler metrics. arxiv:1803.06801 Google Scholar
Futaki, A. and Ono, H., Volume minimization and conformally Kähler, Einstein–Maxwell geometry . J. Math. Soc. Japan 70(2018), 4, 14931521. https://doi.org/10.2969/jmsj/77837783 Google Scholar
Karki, M. B. and Thompson, G., Four-dimensional Einstein Lie groups . Differ. Geom. Dyn. Syst. 18(2016), 4357.Google Scholar
Koca, C. and Tønnesen-Friedman, C. W., Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces . Ann. Global Anal. Geom. 50(2016), no. 1, 2946. https://doi.org/10.1007/s10455-016-9499-z Google Scholar
Lahdili, A., Conformally Kähler, Einstein–Maxwell metrics and boundedness of the modified Mabuchi-functional. arxiv:1710.00235 Google Scholar
Lahdili, A., Automorphisms and deformations of conformally Kähler, Einstein–Maxwell metrics . J. Geom. Anal. 29(2019), no. 1, 542568. https://doi.org/10.1007/s12220-018-0010-x Google Scholar
LeBrun, C., The Einstein–Maxwell equations, extremal Kähler metrics, and Seiberg–Witten theory . In: The many facets of geometry . Oxford University Press, Oxford, 2010, pp. 1733. https://doi.org/10.1093/acprof:oso/9780199534920.003.0003 Google Scholar
LeBrun, C., The Einstein–Maxwell equations, Kähler metrics, and Hermitian geometry . J. Geom. Phys. 91(2015), 163171. https://doi.org/10.1016/j.geomphys.2015.01.009 Google Scholar
LeBrun, C., The Einstein–Maxwell equations and conformally Kähler geometry . Comm. Math. Phys. 344(2016), no. 2, 621653. https://doi.org/10.1007/s00220-015-2568-5 Google Scholar
Mubarakzjanov, G. M., Classification of real structures of Lie algebras of fifth order . Izv. Vysš. Učebn. Zaved. Matematika 1963(1963), no. 3 (34), 99106.Google Scholar
Patera, J., Sharp, R. T., and Winternitz, P., Invariants of real low dimension Lie algebras . J. Math. Phys. 17(1976), 986994. https://doi.org/10.1063/1.522992 Google Scholar
Shao, H., Compactness and rigidity of Kähler surfaces with constant scalar curvature. arxiv:1304.0853 Google Scholar
Yamabe, H., On a deformation of Riemannian structures on compact manifolds . Osaka Math. J. 12(1960), 2137.Google Scholar