The space of Kähler metrics. II.

*(English)*Zbl 1067.58010Summary: This paper, the second of a series [part I, the second author, ibid. 56, No. 2, 189–234 (2000; Zbl 1041.58003)], 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, who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series, 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 \(C^{1,1}\). This partially verifies two conjectures of Donaldson 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 \(M\).

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 \(M\).

##### MSC:

58D27 | Moduli problems for differential geometric structures |

32Q15 | Kähler manifolds |

32W20 | Complex Monge-Ampère operators |

58E11 | Critical metrics |

58D17 | Manifolds of metrics (especially Riemannian) |