scholarly journals Multiplier Hermitian structures on Kähler manifolds

2003 ◽  
Vol 170 ◽  
pp. 73-115 ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Systematic Study ◽  
Einstein Metrics ◽  
Group Actions ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fano Manifold ◽  
Hermitian Manifolds ◽  
Kähler Einstein Metrics ◽  
Hermitian Structures

AbstractThe main purpose of this paper is to make a systematic study of a special type of conformally Kähler manifolds, called multiplier Hermitian manifolds, which we often encounter in the study of Hamiltonian holomorphic group actions on Kähler manifolds. In particular, we obtain a multiplier Hermitian analogue of Myers’ Theorem on diameter bounds with an application (see [M5]) to the uniquness up to biholomorphisms of the “Kähler-Einstein metrics” in the sense of [M1] on a given Fano manifold with nonvanishing Futaki character.

scholarly journals Geometric estimates for complex Monge–Ampère equations

2020 ◽  
Vol 2020 (765) ◽  
pp. 69-99 ◽  
Author(s):  
Xin Fu ◽  
Bin Guo ◽  
Jian Song
Keyword(s):  
Einstein Metrics ◽  
Kähler Manifolds ◽  
Singular Solutions ◽  
Kahler Manifolds ◽  
Diameter Estimate ◽  
Uniform Diameter ◽  
Kähler Einstein Metrics ◽  
Geometric Regularity ◽  
Monge Ampere ◽  
Geometric Estimates

AbstractWe prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler manifolds of nonnegative Kodaira dimensions.

scholarly journals DEGENERATE COMPLEX MONGE–AMPÈRE EQUATIONS OVER COMPACT KÄHLER MANIFOLDS

2010 ◽  
Vol 21 (03) ◽  
pp. 357-405 ◽  
Author(s):  
JEAN-PIERRE DEMAILLY ◽  
NEFTON PALI
Keyword(s):  
Existence And Uniqueness ◽  
General Type ◽  
Einstein Metrics ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Sheaf ◽  
Kähler Einstein Metrics ◽  
Monge Ampere

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge–Ampère equations, and investigate their regularity. These types of equations are precisely what is needed in order to construct Kähler–Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähler–Einstein metrics over varieties of general type.

scholarly journals DEGENERATION OF KÄHLER–EINSTEIN MANIFOLDS I: THE NORMAL CROSSING CASE

2004 ◽  
Vol 06 (02) ◽  
pp. 301-313
Author(s):  
WEI-DONG RUAN
Keyword(s):  
Einstein Metrics ◽  
Total Space ◽  
Kähler Manifolds ◽  
Einstein Metric ◽  
Kahler Manifolds ◽  
Einstein Manifolds ◽  
Normal Crossing ◽  
Smooth Part ◽  
Kähler Einstein Metrics

In this paper we prove that the Kähler–Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles converge in the sense of Cheeger–Gromov to the complete Kähler–Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil–Peterson metric in this case.

scholarly journals Coupled Kähler–Einstein Metrics

2018 ◽  
Vol 2019 (21) ◽  
pp. 6765-6796 ◽  
Author(s):  
Jakob Hultgren ◽  
D Witt Nyström
Keyword(s):  
Stability Condition ◽  
Existence And Uniqueness ◽  
Einstein Metrics ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Existence And Uniqueness Results ◽  
Algebraic Stability ◽  
Canonical Metrics ◽  
Uniqueness Results ◽  
Kähler Einstein Metrics

Abstract We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler–Einstein metrics, generalizing Kähler–Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler–Einstein Fano. In the Fano case, we also prove that existence of coupled Kähler–Einstein metrics imply a certain algebraic stability condition, generalizing K-polystability.

ON HARMONIC AND PSEUDOHARMONIC MAPS FROM PSEUDO-HERMITIAN MANIFOLDS

2017 ◽  
Vol 234 ◽  
pp. 170-210 ◽  
Author(s):  
TIAN CHONG ◽  
YUXIN DONG ◽  
YIBIN REN ◽  
GUILIN YANG
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Rigidity Results ◽  
Hermitian Manifolds

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds into Riemannian manifolds or Kähler manifolds. Some foliated results, pluriharmonicity and Siu–Sampson type results are established for both harmonic maps and pseudoharmonic maps.

scholarly journals Ricci iterations on Kähler classes

2009 ◽  
Vol 8 (4) ◽  
pp. 743-768 ◽  
Author(s):  
Julien Keller
Keyword(s):  
Iterative Procedure ◽  
Einstein Metrics ◽  
Einstein Manifold ◽  
Periodic Points ◽  
Dynamical Behaviour ◽  
Iteration Scheme ◽  
Fano Manifold ◽  
Ricci Operator ◽  
Finite Dimensional ◽  
Kähler Einstein Metrics

AbstractIn this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics of a Fano manifold. Nadel has defined an iteration scheme given by the Ricci operator and asked whether it has some non-trivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of its iterates for a Fano Kähler–Einstein manifold. Then we define a finite-dimensional procedure to give an approximation of Kähler–Einstein metrics using this iterative procedure and apply it on ℂℙ2 blown up in three points.

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