The K\"ahler-Ricci flow on Fano manifolds
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).
Keywords
Cite
@article{arxiv.1212.6227,
title = {The K\"ahler-Ricci flow on Fano manifolds},
author = {Huai-Dong Cao},
journal= {arXiv preprint arXiv:1212.6227},
year = {2024}
}
Comments
v.2: corrected a number of typos and added the proof of Theorem 2.3 on preserving positive orthogonal bisectional curvature. To appear as a book chapter in An Introduction to the K\"ahler-Ricci Flow, Lecture Notes in Mathematics, vol. 2086, Springer, 2013