Donaldson conjectured [14] that the space of Kähler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in each Kähler class if the first Chern class is either strictly negative or 0. Furthermore, if C1 ≤ 0, the constant scalar curvature metric realizes the global minimum of Mabuchi energy functional; thus it provides a new obstruction for the existence of constant curvature metric: if the infimum of Mabuchi energy (taken over all metrics in a fixed Kähler class) isn't bounded from below, then there doesn't exist a constant curvature metric. This extends the work of Mabuchi and Bando[3]: they showed that Mabuchi energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class. * Research was supported partially by NSF postdoctoral fellowship. * Around the same time with Mabuchi's work, Bourguignon J. P. has worked on something similar in a related subject [6].† Here we mean the mixed second derivatives is uniformly bounded. See theorem 3 in section 3 for details. ‡ The sufficient part of this result was proved in [12]. § Tian inform us that he [39] has conjectured that constant scalar curvature metrics exist if and only if Mabuchi functional is proper. * * In [14], Donaldson provided a formal proof to this proposition after assuming the existence of a smooth geodesic between any two metrics. Our proof follows his idea closely.

This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.

This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2 π \pi . We also put all our technical results together to complete the proof of the main theorem.