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We explain in this paper that for a certain completions of C bundles the existence of the Calabi extremal metrics are the same as the geodesic stability and prove the exponential C convergence of the modified Calabi flow whenever the extremal metric exists, i.e., when the Kähler class is geodesically stable, for those cases with hypersurface ends. In particular, we solved the problem of the convergence of the modified Calabi flow on those almost homogeneous manifolds with two hypersurface ends in [Gu2]. As a byproduct, we also found a family of Kähler metrics called extremalsoliton metrics which interpolates the extremal metrics and the generalized quasi-einstein metrics. We also proved that the existence of these metrics on compact almost homogeneous manifolds of two ends. For the completions of the C bundles we considered in this paper, we found a functional which 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 respect to this functional.

Extremal solitons and exponential C∞convergence of the modified Calabi flow on certain ℂP1Bundles

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles