Einstein Kähler Metrics of Negative Ricci Curvature on Open Kähler Manifolds

2018 ◽  
Author(s):  
Shigetoshi Bando
Keyword(s):  
Ricci Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics

scholarly journals Astheno–Kähler and Balanced Structures on Fibrations

2017 ◽  
Vol 2019 (22) ◽  
pp. 7093-7117 ◽  
Author(s):  
Anna Fino ◽  
Gueo Grantcharov ◽  
Luigi Vezzoni
Keyword(s):  
Homogeneous Spaces ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Hermitian Metrics ◽  
Kahler Metrics ◽  
Balanced Metrics ◽  
Invariant Volume ◽  
Balanced Metric ◽  
First Chern Class

Abstract We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, strong Kähler with torsion (SKT), and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then we provide a construction of astheno-Kähler structures on torus bundles over Kähler manifolds leading to new examples. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metric, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups SU(3) and G2 admit SKT and astheno-Kähler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space M with invariant volume admits a balanced metric, then its first Chern class c1(M) does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.

scholarly journals Quaternionic Kähler metrics associated with special Kähler manifolds

2015 ◽  
Vol 92 ◽  
pp. 271-287 ◽  
Author(s):  
D.V. Alekseevsky ◽  
V. Cortés ◽  
M. Dyckmanns ◽  
T. Mohaupt
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Special Kähler Manifolds ◽  
Quaternionic Kähler

scholarly journals A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

2001 ◽  
Vol 12 (07) ◽  
pp. 769-789 ◽  
Author(s):  
VESTISLAV APOSTOLOV ◽  
TEDI DRĂGHICI ◽  
ANDREI MOROIANU
Keyword(s):  
Ricci Tensor ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Complex Dimension ◽  
Negative Eigenvalues ◽  
Kähler Surfaces ◽  
Almost Kähler ◽  
Compact Kähler Manifold

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4.

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