Journal of Differential Geometry

The Space of Kähler Metrics II

E. Calabi and X.X. Chen

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This paper, the second of a series, deals with the function space \mathcal{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 \mathcal{H} is geodesically convex in the sense that, for any two points of \mathcal{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 \mathcal{H} of the "K energy" of V has the property that it strictly decreases the length of all paths in \mathcal{H}, except those induced by one parameter families of holomorphic automorphisms of M.

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J. Differential Geom., Volume 61, Number 2 (2002), 173-193.

First available in Project Euclid: 20 July 2004

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Calabi, E.; Chen, X.X. The Space of Kähler Metrics II. J. Differential Geom. 61 (2002), no. 2, 173--193. doi:10.4310/jdg/1090351383.

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