The Kahler-Ricci Flow on Fano Manifolds

In these lecture notes, we aim at giving an introduction to the Kahler-Ricci flow (KRF) on Fano manifolds. It covers mostly the developments of the KRF in its first 20 years (1984-2003), especially an essentially self-contained exposition of Perelman's uniform estimates on the scalar curvature, the diameter, and the Ricci potential function for the normalized Kahler-Ricci flow (NKRF), including the monotonicity of Perelman's mu-entropy and kappa-noncollapsing theorems for the Ricci flow on compact manifolds. The lecture notes is based on a mini-course on KRF delivered at University of Toulouse III in February 2010, a talk on Perelman's uniform estimates for NKRF at Columbia University's Geometry and Analysis Seminar in Fall 2005, and several conference talks, including "Einstein Manifolds and Beyond" at CIRM (Marseille-Luminy, fall 2007), "Program on Extremal Kahler Metrics and Kahler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic Aspects of Algebraic and Complex Geometry" at CIRM (Marseille-Luminy, spring 2011).