© 2013

An Introduction to the Kähler-Ricci Flow

  • Sebastien Boucksom
  • Philippe Eyssidieux
  • Vincent Guedj

Part of the Lecture Notes in Mathematics book series (LNM, volume 2086)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj
    Pages 1-6
  3. Cyril Imbert, Luis Silvestre
    Pages 7-88
  4. Jian Song, Ben Weinkove
    Pages 89-188
  5. Sébastien Boucksom, Vincent Guedj
    Pages 189-237
  6. Huai-Dong Cao
    Pages 239-297
  7. Back Matter
    Pages 335-336

About this book


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 Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-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 Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation).
As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman’s ideas: the Kähler-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


Complex Monge-Ampère equations Kähler-Ricci flow Minimal Model Program Parabolic equations Perleman's estimates

Editors and affiliations

  • Sebastien Boucksom
    • 1
  • Philippe Eyssidieux
    • 2
  • Vincent Guedj
    • 3
  1. 1.Institut de Mathématiques de JussieuCNRS-Université Pierre et Marie CurieParisFrance
  2. 2.Institut Fourier and Institut Universitaire de FranceUniversité Joseph FourierSaint-Martin d'HèresFrance
  3. 3.Institut de Mathématiques de Toulouse and Institut Universitaire de FranceUniversité Paul SabatierToulouseFrance

Bibliographic information


“This volume comprises contributions to a series of meetings centered around the Kähler-Ricci flow that took place in Toulouse, Marseille, and Luminy in France, as well as in Marrakech, Morocco in 2010 and 2011. … These contributions cover a wide range of the theory and applications of Kähler-Ricci flow and are a welcome addition to the literature on this topic of great current interest in global analysis.” (M. Kunzinger, Monatshefte für Mathematik, 2015)