Vol. 233, No. 1, 2007

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ISSN: 0030-8730
Extremal solitons and exponential C convergence of the modified Calabi flow on certain P1 Bundles

Daniel Guan

Vol. 233 (2007), No. 1, 91–124
Abstract

For a certain class of completions of -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 -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.

Keywords
metrics flow, extremal metrics, complex manifolds, scalar curvature, fiber bundle, homogeneous manifolds
Mathematical Subject Classification 2000
Primary: 53C10, 53C21, 53C26, 53C55, 32F32, 32L05, 32M12, 32Q20
Milestones
Received: 17 April 2006
Revised: 1 December 2006
Accepted: 22 December 2006
Published: 1 November 2007
Authors
Daniel Guan
Department of Mathematics
University of California
Riverside, CA 92521
United States