On any compact Kähler manifold, Mabuchi [16], Semmes [17], and Donaldson [5] introduced a Weil-Peterson type metric in the space of Kähler metrics and proved that it is a formally non-positively curved symmetric space of “noncompact” type. According to [17], the geodesic equation can be transformed into a homogenous complex Monge-Ampere equation. In [5], Donaldson proposed an ambitious program ...
