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 Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

2019 ◽  
Vol 2019 (751) ◽  
pp. 27-89 ◽  
Author(s):  
Robert J. Berman ◽  
Sebastien Boucksom ◽  
Philippe Eyssidieux ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Ricci Flow ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Einstein Metrics ◽  
Discrete Version ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics ◽  
Special Case

AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

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 Ricci Iteration for Coupled Kähler–Einstein Metrics

2019 ◽  
Author(s):  
Ryosuke Takahashi
Keyword(s):  
Dynamical System ◽  
Automorphism Group ◽  
Chern Class ◽  
Einstein Metrics ◽  
Fano Manifolds ◽  
Ricci Operator ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics

Abstract In this paper, we introduce the “coupled Ricci iteration”, a dynamical system related to the Ricci operator and twisted Kähler–Einstein metrics as an approach to the study of coupled Kähler–Einstein (CKE) metrics. For negative 1st Chern class, we prove the smooth convergence of the iteration. For positive 1st Chern class, we also provide a notion of coercivity of the Ding functional and show its equivalence to the existence of CKE metrics. As an application, we prove the smooth convergence of the iteration on CKE Fano manifolds assuming that the automorphism group is discrete.

scholarly journals Remarks on Kähler-Einstein Manifolds

1972 ◽  
Vol 46 ◽  
pp. 161-173 ◽  
Author(s):  
Yozo Matsushima
Keyword(s):  
Einstein Metrics ◽  
Integral Formula ◽  
Curvature Form ◽  
Einstein Metric ◽  
Kähler Metric ◽  
Kähler Einstein Metrics ◽  
Simply Connected ◽  
Kahler Metric ◽  
Complex Projective ◽  
Ricci Form

The main purpose of this note is to characterize a compact Káhler-Einstein manifold in terms of curvature form. The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Káhler metric is the harmonic representative of the curvature class if and only if the Káhler metric is an Einstein metric in the generalized sense (g.s.), that is, if the Ricci form of the metric is parallel. It is well known that a Káhler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kàhler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Káhler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kãhler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s.

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 On the long time behaviour of the conical Kähler–Ricci flows

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiuxiong Chen ◽  
Yuanqi Wang
Keyword(s):  
Einstein Metrics ◽  
Maximal Regularity ◽  
Type Theorem ◽  
Conical Singularities ◽  
Ricci Flows ◽  
Long Time Behaviour ◽  
Long Time ◽  
First Chern Class ◽  
Kähler Einstein Metrics ◽  
Key Steps

Abstract We prove that the conical Kähler–Ricci flows introduced in [11] exist for all time {t\in[0,+\infty)} . These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class {C_{1,\beta}} is negative or zero, the corresponding conical Kähler–Ricci flows converge to Kähler–Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville-type theorem for Kähler–Ricci flat metrics (which are defined over {\mathbb{C}^{n}} ) with conical singularities.

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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