Abstract
The aim of this paper is to study self-similar solutions to the symplectic curvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention on the family of symplectic two- and three-step nilpotent Lie algebras admitting a minimal compatible metric and give a complete classification of these algebras together with their respective metric. Such a classification is given by using our generalization of Nikolayevsky’s nice basis criterion, which, for the convenience of the reader, will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds. By computing the Chern–Ricci operator \(\text {P}\) in each case, we show that the above distinguished metrics define a soliton almost Kähler structure. Many illustrative examples are carefully developed.
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Acknowledgments
The author wishes to express his gratitude to the anonymous referees for comments and valuable criticisms. Edison Alberto Fernández-Culma was fully supported by a CONICET Postdoctoral Fellowship (Argentina).
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Communicated by Eduardo Garcia-Rio.
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Fernández-Culma, E.A. Soliton Almost Kähler Structures on 6-Dimensional Nilmanifolds for the Symplectic Curvature Flow. J Geom Anal 25, 2736–2758 (2015). https://doi.org/10.1007/s12220-014-9534-x
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DOI: https://doi.org/10.1007/s12220-014-9534-x
Keywords
- Symplectic curvature flow
- Self-similar solutions
- Geometric structures on nilmanifolds
- Nilpotent Lie algebras
- Convexity of the moment map
- Nice basis