The Ricci Flow on Some Generalized Wallach Spaces

  • N. A. Abiev
  • A. Arvanitoyeorgos
  • Yu. G. Nikonorov
  • P. Siasos
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 72)

Abstract

We study the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as a special planar dynamical system. All nonsymmetric generalized Wallach spaces can be naturally parametrized by three positive numbers \(a_{1},a_{2},a_{3}\). Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points \((a_{1},a_{2},a_{3})\) in the cube \((0,1/2] \times (0,1/2] \times (0,1/2]\). We also consider in detail some important partial cases.

Keywords

Riemannian metric Einstein metric Generalized Wallach space Ricci flow Ricci curvature Planar dynamical system Real algebraic surface 

Mathematics Subject Classification (2010):

201053C30 53C44 37C10 34C05 14P05 

Notes

Acknowledgements

The authors are indebted to Prof. Yusuke Sakane for useful discussions concerning computational aspects of this project and to Tanya Nikonorova for her help with the graphics. The project was supported in part by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (grant NSh-921.2012.1) and by Federal Target Grant “Scientific and educational personnel of innovative Russia” for 2009–2013 (agreement no. 8206, application no. 2012-1.1-12-000-1003-014).

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • N. A. Abiev
    • 1
  • A. Arvanitoyeorgos
    • 2
  • Yu. G. Nikonorov
    • 3
  • P. Siasos
    • 2
  1. 1.Taraz State University named after M.Kh. DulatyTarazKazakhstan
  2. 2.Department of MathematicsUniversity of PatrasRionGreece
  3. 3.South Mathematical Institute of Vladikavkaz Scientific Centre of the Russian Academy of SciencesVladikavkazRussia

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