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 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 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 Characterizing the complex hyperbolic space by Kähler homogeneous structures

2000 ◽  
Vol 128 (1) ◽  
pp. 87-94 ◽  
Author(s):  
P. M. GADEA ◽  
A. MONTESINOS AMILIBIA ◽  
J. MUÑOZ MASQUÉ
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Complex Hyperbolic Space ◽  
Homogeneous Structure ◽  
Holomorphic Sectional Curvature ◽  
Bergman Metric ◽  
Negative Constant ◽  
Homogeneous Structures ◽  
Simply Connected

The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

Sign in / Sign up

Export Citation Format

Share Document