Ricci curvature and einstein metrics

1981 ◽  
pp. 42-63 ◽  
Author(s):  
Jean Pierre Bourguignon
Keyword(s):  
Ricci Curvature ◽  
Einstein Metrics

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

2018 ◽  
Vol 62 (3) ◽  
pp. 509-523
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Constant Sectional Curvature ◽  
Einstein Metric ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Order Ordinary Differential Equation ◽  
Canonical Metric ◽  
Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

scholarly journals On a class of Einstein Finsler metrics

2014 ◽  
Vol 25 (04) ◽  
pp. 1450030 ◽  
Author(s):  
Zhongmin Shen ◽  
Changtao Yu
Keyword(s):  
Large Class ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Sufficient Conditions ◽  
Riemannian Metric ◽  
Finsler Metrics

In this paper, we study Finsler metrics expressed in terms of a Riemannian metric, an 1-form, and its norm. We find equations which are sufficient conditions for such Finsler metrics to have constant Ricci curvature. Using certain transformations, we successfully solve these equations and hence construct a large class of Einstein metrics.

Ricci curvature in Kähler geometry

2017 ◽  
Vol 28 (09) ◽  
pp. 1740009
Author(s):  
Song Sun
Keyword(s):  
Ricci Curvature ◽  
Einstein Metrics ◽  
Complex Analysis ◽  
Einstein Metric ◽  
Tangent Cones ◽  
Kahler Geometry ◽  
Fano Manifolds ◽  
Winter School ◽  
Kähler Einstein Metrics ◽  
Hausdorff Limits

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

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.

scholarly journals Convergence results for two Kähler–Ricci flows

2014 ◽  
Vol 25 (09) ◽  
pp. 1450084
Author(s):  
Zhou Zhang
Keyword(s):  
Ricci Curvature ◽  
Einstein Metrics ◽  
Curvature Form ◽  
Volume Form ◽  
Ricci Flows ◽  
Convergence Results ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics

In this note, we provide some general discussion on the two main versions in the study of Kähler–Ricci flows over closed manifolds, aiming at smooth convergence to the corresponding Kähler–Einstein metrics with assumptions on the volume form and Ricci curvature form along the flow.

scholarly journals Greatest lower bounds on the Ricci curvature of Fano manifolds

2010 ◽  
Vol 147 (1) ◽  
pp. 319-331 ◽  
Author(s):  
Gábor Székelyhidi
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Einstein Metrics ◽  
Upper Bounds ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Existence Time ◽  
Kähler Einstein Metrics ◽  
Bounded Below ◽  
Kahler Metric

AbstractOn a Fano manifoldMwe study the supremum of the possibletsuch that there is a Kähler metricω∈c1(M) with Ricci curvature bounded below byt. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that onP2blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

scholarly journals Rigidity of gradient Einstein shrinkers

2015 ◽  
Vol 17 (06) ◽  
pp. 1550046 ◽  
Author(s):  
Giovanni Catino ◽  
Lorenzo Mazzieri ◽  
Samuele Mongodi
Keyword(s):  
Differential Geometry ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Global Analysis ◽  
Ricci Solitons ◽  
Global Differential Geometry ◽  
Lecture Notes ◽  
Negative Sectional Curvature

In this paper, we consider a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63] and studied in [H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B27(2) (2006) 121–142] and we classify noncompact gradient shrinkers with bounded non-negative sectional curvature.

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