On the Convergence of the Kahler-Ricci Flow to Singular Kahler Metrics

Two classical problems in Riemannian geometry are the prescribed Ricci curvature problem and the existence of Einstein metrics. In the Kähler setting, the existence and uniqueness of such metrics is equivalent to the existence and uniqueness of solutions to the complex Monge-Ampere equation.

We discuss the existence and uniqueness of solutions to a complex Monge-Ampere equation on a projective variety which yields singular Kähler metrics that are models for the limit of the Kähler-Ricci flow in some natural algebro-geometric settings. We also discuss some consequences for scalar curvature and some other geometric behaviour.