Geometry of K\"ahler Metrics and Foliations by Holomorphic Discs

The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, we will prove the uniqueness of extremal K\"ahler metrics and give an necessary condition for existence of extremal K\"ahler metrics. Further applications will be discussed in our forthcoming papers.