Monge-Ampère Equations on (Para-)Kähler Manifolds: from Characteristic Subspaces to Special Lagrangian Submanifolds

2012 ◽  
Vol 120 (1) ◽  
pp. 3-27 ◽  
Author(s):  
Dmitri Alekseevsky ◽  
Ricardo Alonso-Blanco ◽  
Gianni Manno ◽  
Fabrizio Pugliese
Keyword(s):  
Lagrangian Submanifolds ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Special Lagrangian ◽  
Special Lagrangian Submanifolds ◽  
Monge Ampere

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.

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