This chapter is central to the whole book and perhaps the main reason and justification for it. Much of it is based on a new method for proving the existence of Einstein metrics on odd dimensional manifolds, which is an orbifold version of the Kobayashi bundle construction. The key ingredient comes from the fact that links of isolated hypersurface singularities obtained from weighted homogeneous polynomials admit Sasakian structures. The focus is on the five-dimensional case where several classification results are described. Although a complete classification is perhaps not within reach, the material of this chapter gives a really good grasp of Sasaki-Einstein geometry in dimension five. One of the more important results of this chapter describes the plethora of Sasaki-Einstein metrics and their moduli that naturally occur, in particular, on many odd dimensional homotopy spheres, including exotic spheres. Toric Sasaki-Einstein manifolds are also discussed in some detail as they provide a unique source of irregular Sasaki-Einstein structures. Other important topics include extremal Sasaki metrics, Sasaki-Futaki character, and the Bishop and Lichnerowicz obstructions. The chapter ends with a brief discussion of Sasakian-η-Einsteinmetrics.
Keywords: Bishop and Lichnerowicz obstructions, exotic spheres, extremal Sasaki metrics, Kobayashi orbibundle, Matsushima-Lichnerowicz obstruction, moduli, Sasaki-Einstein metrics, Sasaki-Futaki character, Sasaki cone, Sasaki fi-Einstein metrics
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