Author’s abstract: This is the second part of the author’s paper [Pac. J. Appl. Math. 3, No. 1–2, 43–71 (2011; Zbl 1293.53054)] on the existence of Kähler Einstein metrics of the general type I almost homogeneous manifolds of cohomogeneity one. We actually extend all results in [the author, Int. J. Math. 14, No. 3, 259–287 (2003; Zbl 1048.32014)] to the type I cases. We also prove the existence of smooth geodesics connecting any two given metrics in the Mabuchi moduli space of Kähler metrics, which leads to the uniqueness of our Kähler metrics with constant scalar curvatures if they exist. We obtain a lot of new Kähler-Einstein manifolds as well as Fano manifolds not admitting Kähler-Einstein metrics. Furthermore, we also deal with the cases where a higher codimensional end occurs, then we obtain more Kähler-Einstein manifolds as well as Fano manifolds not admitting Kähler-Einstein metrics. As an offshoot, we are able to classify compact Kähler manifolds which are almost homogeneous of cohomogeneity one with a higher codimensional end. Applying our results to the canonical circle bundles we also obtain Sasakian manifolds with or without Sasakian-Einstein metrics. That also give some open Calabi-Yau manifolds.
For Part I see [the author, ibid. 3, No. 1–2, 43–71 (2011; Zbl 1293.53054)].