Weak solutions to the complex Monge-Ampere equation

摘要

Canonical K?hler metrics, such as Ricci-flat or Kahler-Einstein, are obtained via solving the complex Monge-Ampère equation. The famous Calabi-Yau theorem asserts the existence and regularity of solutions to this equation on compact K?hler manifolds for smooth data. In this note we shall present methods, based on pluripotential theory, which yield the results on the existence and stability of the weak solutions of the Monge-Ampère equation for possibly degenerate, non-smooth right hand side. Those weak solutions have also interesting applications in geometry. They lead to canonical metrics with singularities, which may occur as the limits of the K?hler-Ricci flow or the limits of families of Calabi-Yau metrics when the Kahler class hits the boundary of the K?hler cone.