scholarly journals Numerical Approximations to Extremal Toric Kähler Metrics with Arbitrary Kähler Class

2017 ◽  
Vol 60 (4) ◽  
pp. 893-910
Author(s):  
Stuart James Hall ◽  
Thomas Murphy
Keyword(s):  
Ricci Flow ◽  
Small Eigenvalue ◽  
Einstein Metric ◽  
Numerical Approximations ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Numerical Evidence ◽  
Extremal Metric ◽  
New Algorithms

AbstractWe develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

scholarly journals Complete Kähler–Einstein Metric on Stein Manifolds With Negative Curvature

2020 ◽  
Author(s):  
Man-Chun Lee
Keyword(s):  
Sectional Curvature ◽  
Ricci Flow ◽  
Negative Curvature ◽  
Einstein Metric ◽  
Holomorphic Sectional Curvature ◽  
Kähler Metrics ◽  
Stein Manifolds ◽  
Kahler Metrics ◽  
Negative Constant

Abstract We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow.

scholarly journals Ricci flow and the metric completion of the space of Kähler metrics

2013 ◽  
Vol 135 (6) ◽  
pp. 1477-1505 ◽  
Author(s):  
Brian Clarke ◽  
Yanir A. Rubinstein
Keyword(s):  
Ricci Flow ◽  
Kähler Metrics ◽  
Kahler Metrics

scholarly journals Remarks on extremal Kähler metrics on ruled manifolds

1992 ◽  
Vol 126 ◽  
pp. 89-101 ◽  
Author(s):  
Akira Fujiki
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Metric ◽  
Kähler Metrics ◽  
Algebraic Subgroup ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

scholarly journals The nondegenerate generalized Kähler Calabi–Yau problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Vestislav Apostolov ◽  
Jeffrey Streets
Keyword(s):  
Weak Convergence ◽  
Ricci Flow ◽  
Kahler Geometry ◽  
Kähler Metrics ◽  
Hamiltonian Action ◽  
Deformation Spaces ◽  
Kahler Metrics ◽  
Generalized Kähler Geometry ◽  
Kähler Structures ◽  
The Given

Abstract We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri’s Calabi–Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeroes of a moment map associated to a Hamiltonian action and finding a Kempf–Ness functional. Lastly we indicate the naturality of generalized Kähler–Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf–Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.

scholarly journals Geometry and topology of the space of Kähler metrics on singular varieties

2018 ◽  
Vol 154 (8) ◽  
pp. 1593-1632 ◽  
Author(s):  
Eleonora Di Nezza ◽  
Vincent Guedj
Keyword(s):  
Einstein Metrics ◽  
Normal Space ◽  
Fano Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Metric Properties ◽  
Singular Varieties ◽  
Underlying Space ◽  
And Topology ◽  
Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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