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The Kähler–Ricci Flow on Fano Manifolds

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An Introduction to the Kähler-Ricci Flow

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

Abstract

In these lecture notes, we aim at giving an introduction to the Kähler–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 Kähler–Ricci flow (NKRF), including the monotonicity of Perelman’s μ-entropy and κ-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 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).

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Notes

  1. 1.

    My work was carried out at Columbia University in early 1990s.

  2. 2.

    Perelman also used a similar argument in proving his uniform diameter estimate for the NKRF, see the proof of Claim 6.1 in Sect. 5.6.

  3. 3.

    Perelman’s private lecture was attended by a very small audience, including this author and the authors of [SeT08].

  4. 4.

    Theorems 5.7.5 and 5.7.6 were observed by Hamilton and the author during the IPAM conference “Workshop on Geometric Flows: Theory and Computation” in February, 2004.

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Acknowledgements

This article 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). This article also served as the lecture notes by the author for a graduate course at Lehigh University in spring 2012, as well as a short course at the Mathematical Sciences Center of Tsinghua University in May, 2012. I would like to thank Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi for inviting me to give the mini-course in Toulouse, and especially Vincent Guedj for inviting me to write up the notes for a special volume. I also wish to thank the participants in my courses, especially Qiang Chen, Xin Cui, Chenxu He, Xiaofeng Sun, Yingying Zhang and Meng Zhu, for their helpful suggestions. Finally, I would like to take this opportunity to express my deep gratitude to Professors E. Calabi, R. Hamilton, and S.-T. Yau for teaching me the Kähler geometry, the Ricci flow, and geometric analysis over the years. Partially supported by NSF grant DMS-0909581.

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  1. Department of Mathematics, University of Macau, Macau, China

    Huai-Dong Cao

  2. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    Huai-Dong Cao

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  1. Institut de Mathématiques de Jussieu, CNRS-Université Pierre et Marie Curie, Paris, France

    Sebastien Boucksom

  2. Institut Fourier and Institut Universitaire de France, Université Joseph Fourier, Saint-Martin d'Hères, France

    Philippe Eyssidieux

  3. Institut de Mathématiques de Toulouse and Institut Universitaire de France, Université Paul Sabatier, Toulouse, France

    Vincent Guedj

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Cao, HD. (2013). The Kähler–Ricci Flow on Fano Manifolds. In: Boucksom, S., Eyssidieux, P., Guedj, V. (eds) An Introduction to the Kähler-Ricci Flow. Lecture Notes in Mathematics, vol 2086. Springer, Cham. https://doi.org/10.1007/978-3-319-00819-6_5

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