We will show that constant scalar curvature Kahler(cscK) metric "adjacent" to a given Kahler class is unique up to isomorphism. This generalizes the previous uniqueness theorems of Chen-Tian and Donaldson, where the Kahler calss admits a cscK metric. This is also an infinite dimensional analogue of the Kempf-Ness theorem in G.I.T, which states that the poly-stable orbit is unique in the closure of a semi-stable orbit. In this talk we shall emphasize this analogue, as well as the metric geometry of the space of Kahler metrics in a fixed K\"ahler class. Joint with Xiuxiong Chen.