Donaldson conjectured  that the space of Kähler metrics is geodesically convex by smooth geodesics 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 metrics: realizes the global minimum of the Mabuchi K energy functional; thus it provides a new obstruction for the existence of constant curvature metrics: if the infimum of the K energy (taken over all metrics in a fixed Kähler class) is not bounded from below, then there does not exist a constant curvature metric. This extends the work of Mabuchi and Bando : they showed that K energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class.
"The Space of Kähler Metrics." J. Differential Geom. 56 (2) 189 - 234, October, 2000. https://doi.org/10.4310/jdg/1090347643