Einstein–Kähler Metrics with Positive Ricci Curvature

Futaki, Akito; Mabuchi, Toshiki; Sakane, Yusuke

doi:10.2969/aspm/01820011

Cited by 9 publications

(3 citation statements)

References 63 publications

(90 reference statements)

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“…cit., pp. [116][117][118]. In fact, as Eells-Sampson observe, when f is an immersion, (7.4) can also be proved using the Gauss equations.…”

Section: Uniformity Of the Metric Imentioning

confidence: 99%

Smooth and Singular Kähler–Einstein Metrics

Rubinstein¹

2014

Geometric and Spectral Analysis

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Smooth Kähler-Einstein metrics have been studied for the past 80 years. More recently, singular Kähler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a survey of some of these developments.The Kähler-Einstein (KE) equation is among the oldest fully nonlinear equations in modern geometry. A wide array of tools have been developed or applied towards its understanding, ranging from Riemannian geometry, PDE, pluripotential theory, several complex variables, microlocal analysis, algebraic geometry, probability, convex analysis, and more. The interested reader is referred to the numerous existing surveys on related topics

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“…cit., pp. [116][117][118]. In fact, as Eells-Sampson observe, when f is an immersion, (7.4) can also be proved using the Gauss equations.…”

Section: Uniformity Of the Metric Imentioning

confidence: 99%

Smooth and Singular Kähler–Einstein Metrics

Rubinstein¹

2014

Geometric and Spectral Analysis

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“…Lemma 2.6 [Futaki et al 1990;Mabuchi 1987;Guan 1995a, Lemma 3]. We can regard U as a moment map corresponding to (g,J H ) and g τ as the symplectic reduction ofg at U (τ ).…”

Section: Existence Of the Extremal Solitons On Certain Completions Ofmentioning

confidence: 99%

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Guan¹

2007

Pacific J. Math.

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For a certain class of completions of ‫ރ‬ * -bundles, we show that the existence of Calabi extremal metrics is equivalent to geodesic stability of the Kähler class, and we prove the exponential C ∞ convergence of the modified Calabi flow whenever the extremal metric exists, assuming that the manifold has hypersurface ends. In particular, we solve the problem of convergence of the modified Calabi flow on the almost homogeneous manifolds with two hypersurface ends which we dealt with in a 1995 Transactions paper. As a byproduct, we found a family of Kähler metrics, called extremal soliton metrics, interpolating the extremal metrics and the generalized quasiEinstein metrics. We also proved the existence of these metrics on compact almost homogeneous manifolds of two ends. For the completions of the ‫ރ‬ * -bundles we consider in this paper, we define what we call the generalized Mabuchi functional; the existence of extremal soliton metrics on these manifolds is again equivalent to the geodesic stability of the Kähler class with respect to this functional.

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“…Proof of Corollary 1. First of all, by [FMS90] (see also [Fut83] and [WAN91]) the Futaki invariant of X is non-zero, hence X does not admit a Kähler-Einstein metrics. To prove the rest of the corollary, we fix a (C * ) 4 -action on X in the following way: Consider the standard embeddings of O P 2 (−1) and O P 1 (−1) in to C 3 × P 2 and C 2 × P 1 respectively:…”

Section: Proof Of Corollarymentioning

confidence: 99%

Coupled Kähler–Ricci solitons on toric Fano manifolds

Hultgren

2019

Analysis & PDE

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We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the Kähler-Ricci flow.

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Einstein–Kähler Metrics with Positive Ricci Curvature

Cited by 9 publications

References 63 publications

Smooth and Singular Kähler–Einstein Metrics

Smooth and Singular Kähler–Einstein Metrics

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Coupled Kähler–Ricci solitons on toric Fano manifolds

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Einstein–Kähler Metrics with Positive Ricci Curvature

Cited by 9 publications

References 63 publications

Smooth and Singular Kähler–Einstein Metrics

Smooth and Singular Kähler–Einstein Metrics

Extremal solitons and exponential C∞convergence of the modified Calabi flow on certain ℂP1Bundles

Coupled Kähler–Ricci solitons on toric Fano manifolds

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Product

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Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles