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The Sasaki Cone and Extremal Sasakian Metrics

  • Charles P. Boyer
  • Krzysztof  Galicki
  • Santiago R. Simanca
Part of the Progress in Mathematics book series (PM, volume 271)

Abstract

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone and use its critical points to single out the strongly extremal Reeb vector fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest L 2-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five-dimensional case.

Keywords

Scalar Curvature Maximal Torus Einstein Metrics Constant Scalar Curvature Sasakian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media LLC 2009

Authors and Affiliations

  • Charles P. Boyer
    • 1
  • Krzysztof  Galicki
    • 1
  • Santiago R. Simanca
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA

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