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

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.

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The Kähler–Ricci flow around complete bounded curvature Kähler metrics

Transactions of the American Mathematical Society ◽

10.1090/tran/8015 ◽

2020 ◽

Vol 373 (5) ◽

pp. 3627-3647 ◽

Cited By ~ 2

Author(s):

Albert Chau ◽

Man-Chun Lee

Keyword(s):

Ricci Flow ◽

Kähler Metrics ◽

Bounded Curvature ◽

Kahler Metrics

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Numerical Approximations to Extremal Toric Kähler Metrics with Arbitrary Kähler Class

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091516000444 ◽

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.

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The nondegenerate generalized Kähler Calabi–Yau problem

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0016 ◽

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.

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

Compositio Mathematica ◽

10.1112/s0010437x18007170 ◽

2018 ◽

Vol 154 (8) ◽

pp. 1593-1632 ◽

Cited By ~ 3

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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