The conical complex Monge–Ampère equations on Kähler manifolds

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.

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DEGENERATE COMPLEX MONGE–AMPÈRE EQUATIONS OVER COMPACT KÄHLER MANIFOLDS

International Journal of Mathematics ◽

10.1142/s0129167x10006070 ◽

2010 ◽

Vol 21 (03) ◽

pp. 357-405 ◽

Cited By ~ 21

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.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-012-7 ◽

2010 ◽

Vol 62 (1) ◽

pp. 218-239 ◽

Cited By ~ 1

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.

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Hölder continuous potentials on manifolds with partially positive curvature

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748010000113 ◽

2010 ◽

Vol 9 (4) ◽

pp. 705-718 ◽

Cited By ~ 2

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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