TYPE I ALMOST-HOMOGENEOUS MANIFOLDS OF COHOMOGENEITY ONE-I

• Published in: Pacific journal of applied mathematics Vol. 3; no. 1/2; p. 45
• Main Author: Guan, Daniel
• Format: Journal Article
• Language: English
• Published: Hauppauge Nova Science Publishers, Inc 01.01.2011
• Subjects: Bundles; Constants; Curvature; Differential equations; Equivalence; Geometry; Lie groups; Manifolds; Mathematical analysis; Nonlinear equations; Scalars; Topological manifolds; Uniqueness
• ISSN: 1941-3963

In this paper, we generalize our results in [GC, Gu4, 5] on the existence of Kähler metrics with constant scalar curvatures to the general type I almost homogeneous manifolds of cohomogeneity one. We actually carry out all the results in [Gu5] to the type I cases. We prove that the existence of Kähler metrics with constant scalar curvatures is equivalent to the negativity of an integral, and is also equivalent to the geodesic stability. We also prove the existence of smooth geodesic connecting any two given metrics on the Mabuchi moduli space of Kähler metrics, which leads to the uniqueness of our Kähler metrics with constant scalar curvatures if they exist. The similar proofs of the results other than the existence of Kähler metrics with constant scalar curvatures for the type II cases are more complicated and will be done in [Gu6]. In particular, we also deal with the existence of Kähler-Einstein metrics on these manifolds and obtain a lot of new Kähler-Einstein manifolds as well as Fano manifolds without KählerEinstein metrics. With applying our results to the canonical circle bundles we also obtain Sasakian manifolds with or without Sasakian-Einstein metrics.