For a certain class of completions of ‫ރ‬ * -bundles, we show that the existence of Calabi extremal metrics is equivalent to geodesic stability of the Kähler class, and we prove the exponential C ∞ convergence of the modified Calabi flow whenever the extremal metric exists, assuming that the manifold has hypersurface ends. In particular, we solve the problem of convergence of the modified Calabi flow on the almost homogeneous manifolds with two hypersurface ends which we dealt with in a 1995 Transactions paper. As a byproduct, we found a family of Kähler metrics, called extremal soliton metrics, interpolating the extremal metrics and the generalized quasiEinstein metrics. We also proved the existence of these metrics on compact almost homogeneous manifolds of two ends. For the completions of the ‫ރ‬ * -bundles we consider in this paper, we define what we call the generalized Mabuchi functional; the existence of extremal soliton metrics on these manifolds is again equivalent to the geodesic stability of the Kähler class with respect to this functional.

In this paper we prove that on certain manifolds Nn with nonnegative first Chern class the existence of extremal metric in a Kähler class is the same as the stability of the Kähler class. We also obtain many new Kähler classes with extremal metrics, in particular, the Kähler-Einstein metrics for Nn with n > 2. We also compare the problem of finding Calabi extremal metrics with the similar problem of finding Hermitian-Einstein metrics on the holomorphic vector bundles. We explain the geodesic stability and found that the stability for the manifold is much more complicated

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Motivated by Koiso’s work on quasi-Einstein metrics on Fano manifolds, we define (generalized) quasi-Einstein metrics in any Kähler class on any compact complex manifold. It turns out that these metrics are similar to Calabi’s Extremal metrics. Moreover their existence might be studied by a curvature flow in a given Kähler class.