  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.
