Canonical Metrics in Sasakian Geometry

The aim of this thesis is to study the existence problem for canonical Sasakian metrics, primarily Sasaki-Einstein metrics. We are interested in providing both necessary conditions, as well as sufficient conditions for the existence of such metrics.;We establish several sufficient conditions for the existence of Sasaki-Einstein metrics by studying the Sasaki-Ricci flow. In the process, we extend some fundamental results from the study of the Kahler-Ricci flow to the Sasakian setting. This includes finding Sasakian analogues of Perelman's energy and entropy functionals which are monotonic along the Sasaki-Ricci flow. Using these functionals we extend Perelman's deep estimates for the Kahler-Ricci flow to the Sasaki-Ricci flow. Namely, we prove uniform scalar curvature, diameter and non-collapsing estimates along the Sasaki-Ricci flow. We show that these estimates imply a uniform transverse Sobolev inequality. Furthermore, we introduce the sheaf of transverse foliate vector fields, and show that it has a natural, transverse complex structure. We show that the convergence of the flow is intimately related to the space of global transversely holomorphic sections of this sheaf.;We introduce an algebraic obstruction to the existence of constant scalar curvature Sasakian metrics, extending the notion of K-stability for projective varieties.;Finally, we show that, for regular Sasakian manifolds whose quotients are Kahler-Einstein Fano manifolds, the Sasaki-Ricci flow, or equivalently, the Kahler-Ricci flow, converges exponentially fast to a (transversely) Kahler-Einstein metric.