scholarly journals A construction of Poincaré–Einstein metrics of cohomogeneity one on the ball

2019 ◽  
Vol 147 (9) ◽  
pp. 3983-3993
Author(s):  
Yoshihiko Matsumoto
Keyword(s):  
Einstein Metrics ◽  
Cohomogeneity One

scholarly journals Cohomogeneity-one quasi-Einstein metrics

2019 ◽  
Vol 470 (1) ◽  
pp. 201-217
Author(s):  
Timothy Buttsworth
Keyword(s):  
Einstein Metrics ◽  
Cohomogeneity One

Affine compact almost-homogeneous manifolds of cohomogeneity one

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Daniel Guan
Keyword(s):  
Einstein Metrics ◽  
Compact Manifolds ◽  
Complex Manifolds ◽  
Homogeneous Manifolds ◽  
Einstein Manifolds ◽  
Extremal Metrics ◽  
Cohomogeneity One ◽  
Fano Manifolds ◽  
Kähler Einstein Metrics

AbstractThis paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

The ODE System Arising from Cohomogeneity One Einstein Metrics

2021 ◽  
pp. 157-165
Author(s):  
J.-H. Eschenburg ◽  
Mckenzie Y. Wang
Keyword(s):  
Einstein Metrics ◽  
Cohomogeneity One ◽  
Ode System

Kähler-Einstein metrics of cohomogeneity one

1998 ◽  
Vol 312 (3) ◽  
pp. 503-526 ◽  
Author(s):  
Andrew Dancer ◽  
McKenzie Y. Wang
Keyword(s):  
Einstein Metrics ◽  
Cohomogeneity One ◽  
Kähler Einstein Metrics

scholarly journals Diagonalizing cohomogeneity-one Einstein metrics

2009 ◽  
Vol 59 (9) ◽  
pp. 1271-1284 ◽  
Author(s):  
Brandon Dammerman
Keyword(s):  
Einstein Metrics ◽  
Cohomogeneity One

scholarly journals Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

2018 ◽  
Author(s):  
Zhuang-dan Guan ◽  
Pilar Orellana ◽  
Anthony Van
Keyword(s):  
General Type ◽  
Recent Progress ◽  
Einstein Metrics ◽  
Type I ◽  
Homogeneous Manifolds ◽  
Einstein Manifolds ◽  
Sasakian Manifolds ◽  
Cohomogeneity One ◽  
Fano Manifolds ◽  
Circle Bundles

This is the fourth part of [6] on the existence of K¨ahler Einstein metrics of the general type I almost homogeneous manifolds of cohomogeneity one. We actually carry out all the results in [8] to the type I cases. In part II [14], we obtained a lot of new K¨ahler-Einstein manifolds as well as Fano manifolds without K¨ahler-Einstein metrics. In particular, by applying Theorem 15 therein, we have complete results in the Theorems 3 and 4 in that paper. However, we only have some partial results in Theorem 5 there. In this note, we shall give a report of recent progress on the Fano manifolds Nn,m when n > 15 and N′n,m when n > 4. We actually give two nice pictures for these two classes of manifolds. See our Theorems 1 and 2 in the last section. Moreover, we post two conjectures. Once we could solve these two conjectures, the question for these two classes of manifolds would be completely solved. With 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.

scholarly journals Four-dimensional Einstein manifolds with Heisenberg symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha
Keyword(s):  
Heisenberg Group ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Hyperbolic Metric ◽  
Einstein Manifolds ◽  
Cohomogeneity One ◽  
Ricci Flat ◽  
Dimensional Group ◽  
The One ◽  
Group Of Isometries

AbstractWe classify Einstein metrics on $$\mathbb {R}^4$$ R 4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.

Non-existence of cohomogeneity one Einstein metrics

1999 ◽  
Vol 314 (1) ◽  
pp. 109-125 ◽  
Author(s):  
Christoph Böhm
Keyword(s):  
Einstein Metrics ◽  
Cohomogeneity One

scholarly journals Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 2
Author(s):  
Zhuang-Dan Guan ◽  
Pilar Orellana ◽  
Anthony Van
Keyword(s):  
Einstein Metrics ◽  
Type I ◽  
Homogeneous Manifolds ◽  
Einstein Manifolds ◽  
Extremal Metrics ◽  
Cohomogeneity One ◽  
Fano Manifolds ◽  
Circle Bundles ◽  
Kähler Einstein Metrics ◽  
Calabi Extremal Metrics

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.

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