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 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 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.

scholarly journals The General Definition of the Complex Monge–Ampère Operator on Compact Kähler Manifolds

2010 ◽  
Vol 62 (1) ◽  
pp. 218-239 ◽  
Author(s):  
Yang Xing
Keyword(s):  
Convergence Theorem ◽  
Convex Cone ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
General Definition ◽  
Kahler Manifolds ◽  
Plurisubharmonic Functions ◽  
Definition Of ◽  
Compact Kähler Manifold ◽  
Monge Ampere

AbstractWe introduce a wide subclass of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well defined and the convergence theorem is valid. We also prove that is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class.

Hölder continuous potentials on manifolds with partially positive curvature

2010 ◽  
Vol 9 (4) ◽  
pp. 705-718 ◽  
Author(s):  
Sławomir Dinew
Keyword(s):  
Positive Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Hölder Continuous ◽  
Bisectional Curvature ◽  
Right Hand ◽  
Monge Ampere

AbstractIt is proved that solutions of the complex Monge–Ampère equation on compact Kähler manifolds with right hand side in Lp, p > 1, are uniformly Hölder continuous under the assumption on non-negative orthogonal bisectional curvature.

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