The Space of Kaehler metrics

Donaldson conjectured \cite{Dona96} that the space of K\"ahler 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 {\bf each} K\"{a}hler class if the first Chern class is either strictly negative or 0. Furthermore, if $C_1 \leq 0,$ 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 K\"{a}hler class) isn't bounded from below, then there doesn't exist a constant curvature metric. This extends the work of Mabuchi and Bando\cite{Bando87}: they showed that Mabuchi energy bounded from below is a necessary condition for the existence of K\"{a}hler-Einstein metrics in the first Chern class.