Kähler metrics ( II )

This paper, the second of a series, deals with the function space of all smooth Kähler metrics in any given closed complex manifold M in a fixed cohomology class. This function space is equipped with a pre-Hilbert manifold structure introduced by T. Mabuchi [10], where he also showed formally it has non-positive curvature. The previous result of the second author [4] showed that the space is a path length space and it is geodesically convex in the sense that any two points are joined by a unique path, which is always length minimizing and of class C. This already confirms one of Donaldson’s conjecture completely and verifies another one partially (cf. [8]). In the present paper, we show first of all, that the space is, as expected, a path length space of non-positive curvature in the sense of A. D. Alexanderov. A second result is related to the theory of extremal Kähler metrics, namely that the gradient flow of the K energy is strictly length decreasing on all paths except those induced by a path of holomorphic automorphisms of M. This result, in particular, implies that extremal Kähler metric is unique up to holomorphic transformations, provided that Donaldson’s conjecture on the regularity of geodesic is true.