| This thesis is divided into two parts. In the first part we will show that there exists a Kahler-Einstein metric on the open Riemann surfaces which is continuously dependent on the initial value. We then generalize this to the higher dimensional complex manifolds, in particular we point out that the metric constructed by Yau-Cheng, Tian, Kobayashi depends continuously on the initial values. We also proved a general theorem about the existence of Kahler-Einstein metric. In the second part of the thesis we prove a Sobolev-Nirenberg type inequality on the real algebraic set. We believe that this result could be fundamentally important to the study of analytic and geometric properties of the real algebraic set. |
