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

Abstract

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.

Generalizations of Kähler-Ricci solitons on projective bundles

“…In [6], [7], Guan defined and studied Generalized Quasi-Einstein (GQE) Kähler metrics. On compact manifolds, these are Kähler metrics for which the Ricci potential is also a Killing potential.…”

“…This notion includes gradient Ricci solitons as a special case, and is thus a natural object of study (such solitons are called Quasi-Einstein metrics in some Physics references). In [7], GQE metrics are studied in relation to a modified Calabi flow. Finally, like extremal Kähler metrics, GQE metrics generalize the notion of constant scalar curvature (CSC) Kähler metrics.…”

“…Existence of GQE metrics has been demonstrated in [6], [7], and [11] in all Kähler classes of certain manifolds. Here we consider a broader class of spaces, namely projective bundles over local products of CSC Kähler manifolds that are admissible in the sense defined in [2].…”

Affine compact almost-homogeneous manifolds of cohomogeneity one

“…There always exists an extremal metric in any Kähler class. Recently, we generalized this existence result to a family of metrics, which connects the extremal metric [10] and the generalized quasi-einstein metric [9], called the extremal-soliton metrics in [16]. The existence of the extremal-soliton is the same as the geodesic stability with respect to a generalized Mabuchi functional.…”

The Ricci iteration and its applications

“…This flow, first studied by Guan [5], can in turn be considered as a Kähler version of Hamilton's Ricci flow. Our work is motivated by the following conjecture (an analogue may also be posed for the flow):…”