Conformally Kähler geometry and quasi-Einstein metrics

Abstract

We prove that the quasi-Einstein metrics found by Lü, Page and Pope on $\mathbb{C}{P}^1$-bundles over Fano Kähler-Einstein bases are conformally Kähler and that the Kähler class of the conformal metric is a multiple of the first Chern class. A detailed study of the lowest-dimensional example of such metrics on $\mathbb{C}{P}^2♯{\overline{\mathbb{C}P}}^2$ using the methods developed by Abreu and Guillemin for studying toric Kähler metrics is given. Our methods yield, in a unified framework, proofs of the existence of the Page, Koiso-Cao and Lü-Page-Pope metrics on $\mathbb{C}{P}^2♯{\overline{\mathbb{C}P}}^2$. Finally, we investigate the properties that similar quasi-Einstein metrics would have if they also exist on the toric surface $\mathbb{C}{P}^2♯2{\overline{\mathbb{C}P}}^2$.

16 pages

References in corpus (1)

• Metrics of Special Curvature with Symmetry