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The Kähler-Ricci flow on Fano manifolds

Huai-Dong Cao

Abstract

In this lecture notes, we aim at giving an introduction to the Kähler-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 Kähler-Ricci flow (NKRF), including the monotonicity of Perelman's μ-entropy and κ-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 Kähler Metrics and Kähler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic Aspects of Algebraic and Complex Geometry" at CIRM (Marseille - Luminy, spring 2011).

The Kähler-Ricci flow on Fano manifolds

Abstract

In this lecture notes, we aim at giving an introduction to the Kähler-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 Kähler-Ricci flow (NKRF), including the monotonicity of Perelman's μ-entropy and κ-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 Kähler Metrics and Kähler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic Aspects of Algebraic and Complex Geometry" at CIRM (Marseille - Luminy, spring 2011).

Paper Structure

This paper contains 7 sections, 38 theorems, 324 equations.

Table of Contents

  1. Preliminaries
  2. The Kähler-Ricci flow and the normalized Kähler-Ricci flow
  3. The Li-Yau-Hamilton inequalities for KRF
  4. Perelman's $\mu$-entropy and $\kappa$-noncollapsing theorems
  5. Uniform curvature and diameter estimates for NKRF with nonnegative bisectional curvature
  6. Perelman's uniform scalar curvature and diameter estimates for NKRF
  7. Remarks on the formation of singularities in KRF

Key Result

Proposition 2.1

Given any initial Kähler metric $\tilde{g}$ on a compact Kähler manifold $X^n$, KRF (2.1) admits a unique solution $g(t)$ for a short time.

Theorems & Definitions (50)

  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 2.1
  • Lemma 2.4
  • Theorem 2.1: Cao Cao85
  • Lemma 2.5
  • ...and 40 more