Sasaki-Ricci flow and transverse Kähler geometry

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It is the purpose of this project to study in details the transverse Kähler structure on a Sasaki manifold and to prove existence results of transverse Einstein-Sasaki metrics and of Sasaki-Einstein metrics. Of particular interest will be the case of a positive basic first Chern class. Moreover, we consider a new and promising global method which lies in the intersection of complex and real analysis, namely the Sasaki-Ricci flow which can be viewed as the analogue to the Kähler-Ricci flow on the space of Sasakian metrics. A prime motivation for this project is the much studied analogue case of the existence of Kähler-Einstein metrics on Kähler manifolds resp. of the Kähler-Ricci flow on the space of Kähler metrics. In particular we want to solve various questions in the context of Sasakian manifolds, like the existence of invariants, Mabuchi functionals and obstructions for the existence of transverse Einstein-Sasaki metrics. The study of the transverse Kähler cone, of transverse holomorphic vector fields and of Sasaki-Ricci solitons should provide crucial information about the structure of the underlying Sasaki manifold.