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Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

1
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China
2
Department of Mathematics, The University of California at Riverside, Riverside, CA 92521, USA
3
Department of Mathematics and Statistics, California State University at Long Beach, Long Beach, CA 90815, USA
4
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 article belongs to the Special Issue Applications of Differential Geometry)
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Abstract

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 N n , m when n > 15 and N n , m when n > 4 . 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. View Full-Text
Keywords: Kähler manifolds; Einstein metrics; Ricci curvature; fibration; almost-homogeneous; cohomogeneity one; semisimple Lie group; Sasakian–Einstein; Calabi–Yau metrics Kähler manifolds; Einstein metrics; Ricci curvature; fibration; almost-homogeneous; cohomogeneity one; semisimple Lie group; Sasakian–Einstein; Calabi–Yau metrics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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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.

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