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Convergence of the Kähler–Ricci Flow on a Kähler–Einstein Fano Manifold

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An Introduction to the Kähler-Ricci Flow

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

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Abstract

The goal of these notes is to sketch the proof of the following result, due to Perelman and Tian–Zhu: on a Kähler–Einstein Fano manifold with discrete automorphism group, the normalized Kähler–Ricci flow converges smoothly to the unique Kähler–Einstein metric. We also explain an alternative approach due to Berman–Boucksom–Eyssidieux–Guedj–Zeriahi, which only yields weak convergence but also applies to Fano varieties with log terminal singularities.

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Notes

  1. 1.

    This functional seems to have been first explicitly considered by W.Y. Ding in [Ding88, p. 465], hence the chosen terminology.

  2. 2.

    In his seminar talk, Perelman apparently focused on his key estimates and did not say much about the remaining details.

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Acknowledgements

It is a pleasure to thank D.H.Phong for patiently explaining several aspects of the proof of this result.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse and Institut Universitaire de France, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Vincent Guedj

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  1. Vincent Guedj
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Correspondence to Vincent Guedj .

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Editors and Affiliations

  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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Guedj, V. (2013). Convergence of the Kähler–Ricci Flow on a Kähler–Einstein Fano Manifold. 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_6

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