The space of kähler metrics II

This paper, the second of a series, deals with the function space H of all smooth Kähler metrics in any given n-dimensional, closed complex manifold V, these metrics being restricted to a given, fixed, real cohomology class, called a polarization of V . This function space is equipped with a pre- Hilbert metric structure introduced by T. Mabuchi [10], who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series [4], the second author showed that the same space is a path length space. He also proved that H is geodesically convex in the sense that, for any two points of H, there is a unique geodesic path joining them, which is always length minimizing and of class C1,1. This partially verifies two conjectures of Donaldson [8] on the subject. In the present paper, we show first of all, that the space is, as expected, a path length space of nonpositive curvature in the sense of A. D. Aleksandrov. A second result is related to the theory of extremal Kähler metrics, namely that the gradient flow in H of the “K energy” of V has the property that it strictly decreases the length of all paths in H, except those induced by one parameter families of holomorphic automorphisms of M.