度量几何是研究流形结构的基本工具，在很多几何问题中发挥了作用。Kahler Einstein度量的存在性是Kahler几何中的重要问题，最近被田刚和Chen-Donaldson-Sun分别独立解决了，证明中的关键就是田刚提出的partial C^0估计，而对此的证明又离不开Cheeger-Colding-Tian的理论，它刻画了极限空间的奇点结构。 Kahler-Ricci孤立子是Kahler Einstein的自然类似物，它的存在性也是很重要的问题。为了应用紧性理论，我们将Cheeger-Colding-Tian的理论推广到Bakry-Emery曲率有下界的情形，在此基础上我们对Kahler-Ricci孤立子的存在性做了大量基础性的工作。本项目将围绕Kahler几何中和Ricci曲率，Bakry-Emery曲率有关的问题展开，特别是Kahler-Ricci孤立子的存在性问题。

Metric geometry is a powerful tool in the study of the structure of manifolds. It has many applications in geometry. The existence of Kahler Einstein metric is an important problem in Kahler geometry which has been solved recently by Gang Tian and Chen-Donaldson-Sun independently. The partial C^0 estimate is the key point in the proof. Cheeger-Colding-Tian's theory describes the singularity of the limit space which plays an important role in the proof of partial C^0 estimate. Kahler-Ricci soliton is a natural counterpart of Kahler Einstein metric. The existence of Kahler-Ricci soliton is also very important. We need to extend the Cheeger-Colding-Tian's theory to manifolds with Bakry-Emery Ricci curvature bounded from below to apply the compactness theory the problem of exsistence of Kahler-Ricci soliton. In this reseach, I will stduy the geometric proble related with Ricci curvature or Bakry-Emery Ricci curvature in the Kahler geometry, especially the existence of Kahler Ricci soliton.