Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China
Department of Mathematics, The University of California at Riverside, Riverside, CA 92521, USA
Department of Mathematics and Statistics, California State University at Long Beach, Long Beach, CA 90815, USA
Amphastar Pharmaceuticals Inc., 11570 6th Street, Rancho Cucamonga, CA 91730, USA
Author to whom correspondence should be addressed.
Received: 16 November 2018 / Revised: 17 December 2018 / Accepted: 19 December 2018 / Published: 25 December 2018
This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds
. We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.
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Guan, Z.-D.D.; Orellana, P.; Van, A. Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV. Axioms 2019, 8, 2.
Guan Z-DD, Orellana P, Van A. Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV. Axioms. 2019; 8(1):2.
Guan, Zhuang-Dan D.; Orellana, Pilar; Van, Anthony. 2019. "Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV." Axioms 8, no. 1: 2.
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