# The Complex Monge-Ampère Equation and Pluripotential Theory

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*Sławomir Kołodziej*

We collect here results on the existence and stability of weak solutions of complex Monge-Ampére equation proved by applying pluripotential theory methods and obtained in past three decades. First we set the stage introducing basic concepts and theorems of pluripotential theory. Then the Dirichlet problem for the complex Monge-Ampére equation is studied. The main goal is to give possibly detailed description of the nonnegative Borel measures which on the right hand side of the equation give rise to plurisubharmonic solutions satisfying additional requirements such as continuity, boundedness or some weaker ones. In the last part the methods of pluripotential theory are implemented to prove the existence and stability of weak solutions of the complex Monge-Ampére equation on compact Kähler manifolds. This is a generalization of the Calabi-Yau theorem.

#### Table of Contents

# Table of Contents

## The Complex Monge-Ampere Equation and Pluripotential Theory

- Contents vii8 free
- Introduction ix10 free
- Chapter 1. Positive currents and plurisubharmonic functions 112 free
- Chapter 2. Siciak's extremal function and a related capacity 1930
- Chapter 3. The Dirichlet problem for the Monge–Ampere equation with continuous data 2334
- Chapter 4. The Dirichlet problem continued 2940
- Chapter 5. The Monge–Ampère equation for unbounded functions 4152
- Chapter 6. The complex Monge–Ampère equation on a compact Kähler manifold 5162
- Bibliography 6374