A viscosity approach to degenerate complex Monge-Ampère equations

Ahmed Zeriahi

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 4, page 843-913
  • ISSN: 0240-2963

Abstract

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This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows us to prove uniqueness of solutions with prescribed boundary conditions.We will prove a global viscosity comparison principle for degenerate complex Monge-Ampère equations on compact Kähler manifolds and show how to combine Viscosity methods and Pluripotential methods to get “continuous versions" of the Calabi-Yau and Aubin-Yau Theorems in some degenerate situations. In particular we prove the existence of singular Kähler-Einstein metrics with continuous potentials on compact normal Kähler varieties with mild singularities and ample or trivial canonical divisor.

How to cite

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Zeriahi, Ahmed. "A viscosity approach to degenerate complex Monge-Ampère equations." Annales de la faculté des sciences de Toulouse Mathématiques 22.4 (2013): 843-913. <http://eudml.org/doc/275319>.

@article{Zeriahi2013,
abstract = {This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows us to prove uniqueness of solutions with prescribed boundary conditions.We will prove a global viscosity comparison principle for degenerate complex Monge-Ampère equations on compact Kähler manifolds and show how to combine Viscosity methods and Pluripotential methods to get “continuous versions" of the Calabi-Yau and Aubin-Yau Theorems in some degenerate situations. In particular we prove the existence of singular Kähler-Einstein metrics with continuous potentials on compact normal Kähler varieties with mild singularities and ample or trivial canonical divisor.},
author = {Zeriahi, Ahmed},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {complex Monge-Ampère equation; compact Kähler manifolds; viscosity approach},
language = {eng},
month = {6},
number = {4},
pages = {843-913},
publisher = {Université Paul Sabatier, Toulouse},
title = {A viscosity approach to degenerate complex Monge-Ampère equations},
url = {http://eudml.org/doc/275319},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Zeriahi, Ahmed
TI - A viscosity approach to degenerate complex Monge-Ampère equations
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 4
SP - 843
EP - 913
AB - This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows us to prove uniqueness of solutions with prescribed boundary conditions.We will prove a global viscosity comparison principle for degenerate complex Monge-Ampère equations on compact Kähler manifolds and show how to combine Viscosity methods and Pluripotential methods to get “continuous versions" of the Calabi-Yau and Aubin-Yau Theorems in some degenerate situations. In particular we prove the existence of singular Kähler-Einstein metrics with continuous potentials on compact normal Kähler varieties with mild singularities and ample or trivial canonical divisor.
LA - eng
KW - complex Monge-Ampère equation; compact Kähler manifolds; viscosity approach
UR - http://eudml.org/doc/275319
ER -

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