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This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the K hler-Ricci flow and its current state-of-the-art. While several excellent books on K hler-Einstein geometry are available, there have been no such works on the K hler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.

The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman's celebrated proof of the Poincar conjecture. When specialized for K hler manifolds, it becomes the K hler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Amp re equation).
As a spin-off of his breakthrough, G. Perelman proved the convergence of the K hler-Ricci flow on K hler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman's ideas: the K hler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman's surgeries.