The K\"ahler-Ricci flow on Fano manifolds

TL;DR

This paper introduces the K"ahler-Ricci flow on Fano manifolds, detailing its development over twenty years and explaining key estimates and properties like Perelman's uniform bounds and entropy monotonicity.

Contribution

It provides an essentially self-contained exposition of the K"ahler-Ricci flow on Fano manifolds, emphasizing Perelman's estimates and their implications for the flow's behavior.

Findings

Perelman's uniform estimates on scalar curvature, diameter, and Ricci potential.

Monotonicity of Perelman's d-entropy.

k-noncollapsing theorems for Ricci flow.

Abstract

In this lecture notes, we aim at giving an introduction to the K\"ahler-Ricci flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in its first twenty 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 K\"ahler-Ricci flow (NKRF), including the monotonicity of Perelman's \mu-entropy and \kappa-noncollapsing theorems for the Ricci flow on compact manifolds. The 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 K\"ahler Metrics and K\"ahler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic Aspects of Algebraic and Complex Geometry" at CIRM (Marseille - Luminy, spring 2011).