The Oscillation Estimate Of Solutions Of A Family Of Complex Monge-Ampere Equations

In this thesis, we try to find new approaches to explore the existence problem of Kahler-Einstein metric on compact Kahler manifold with C1(M)>0regarding to α-invariant (firstly obtained by Tian [1] in1987). More specifically, we do the oscillation estimate of solutions of a family of complex Monge-Ampere equations (*)t which are introduced for solving the existence problem. These basic ideas are mainly motivated by the result of Tian and Zhu in [4] and the work of Demailly and Pali in [5]. Meanwhile, we obtain some new estimates on a family of complex Monge-Ampere equations related to Kahler-Einstein metric existence.In the first part, we introduce some related concepts such as α-invariant and one main re-sult on the existence problem when α-invariant is bigger than n/(n+1). In the second part, we give our proof of the main result. In the third part, we discuss the case when α-invariant is equal to n/(n+1)after providing a short remark on our former proof. In addition, we try to explain another stronger lemma from Demailly which could also lead to the main result.