Sasaki-Ricci flow and transverse Kähler geometry

本课题的目的是详细研究Sasaki流形上的横向Kähler结构，并证明横向Einstein-Sasaki度量和Sasaki- einstein度量的存在性结果。特别令人感兴趣的将是一个正基本的第一陈氏类的情况。此外，我们考虑了一种新的和有前途的全局方法，它位于复分析和实分析的交叉点，即Sasaki-Ricci流，它可以看作是sasaki度量空间上Kähler-Ricci流的类似物。这个项目的主要动机是研究了Kähler流形上存在Kähler-Einstein度量的模拟情况。在Kähler度量空间上的Kähler-Ricci流。特别是我们想要解决sasaki流形背景下的各种问题，如不变量的存在性，Mabuchi泛函和横向爱因斯坦- sasaki度量存在的障碍。横向Kähler锥、横向全纯向量场和Sasaki- ricci孤子的研究将提供关于底层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.