Sasaki-Ricci flow and transverse Kähler geometry

本项目的目的是详细研究Sasaki流形上的横Kähler结构，并证明横Einstein-Sasaki度量和Sasaki-Einstein度量的存在性结果。特别感兴趣的将是一个积极的基本第一陈类的情况。此外，我们考虑了一个新的和有前途的全球性的方法，这是在复杂的和真实的分析，即Sasaki-Ricci流，可以被看作是类似的Sasakian度量空间上的Kähler-Ricci流的交叉点。这个项目的主要动机是大量研究的类似情况下存在的凯勒-爱因斯坦度量的凯勒流形。Kähler度量空间上的Kähler-Ricci流。特别是，我们要解决各种问题的背景下，Sasaki流形，如存在的不变量，马渊泛函和障碍的存在横向爱因斯坦-Sasaki度量。对横向凯勒锥、横向全纯矢量场和Sasaki-Ricci孤子的研究应该提供有关Sasaki流形结构的重要信息。

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.