For a certain class of
completions of ℂ∗‘-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 quasi-Einstein
metrics. We also proved the existence of these metrics on compact almost
homogeneous manifolds of two ends. For the completions of the ℂ∗‘-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.
