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Concavity of Perelman’s \(\mathcal {W}\)-functional over the space of Kähler potentials

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Abstract

The concavity of Perelman’s \(\mathcal {W}\)-functional over a neighborhood of a Kähler–Ricci soliton inside the space of Kähler potentials is a direct consequence of author’s solution of the variational stability problem for Kähler–Ricci solitons. We provide a new and rather simple proof of this particular fact. This new proof uses in minor part some elementary formulas obtained in our previous work.

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References

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Acknowledgements

I thank the referees for their comments and suggestions on the preliminary version of this manuscript. I also thank all who gave me the opportunity to present the results of [7, 8] to a wide audience. In particular, I thank Professor Jean-Michel Bismut for inviting me to his seminar in 2013, Professor Gang Tian for inviting me to the Princeton “Differential Geometry and Geometric Analysis Seminar” in May 2015, Professor Xiuxiong Chen for inviting me to the Annual Shanks Conference in May 2015 followed by a talk at the USTC, Hefei, June 2015, Professor Jean-Pierre Demailly for inviting me to the workshop ALKAGE in May 2016, and Professor Duong H. Phong for agreeing to be a member of the jury and a referee of my Habilitation Thesis in May 2015.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris Sud, Bâtiment 425, 91405, Orsay, France

    Nefton Pali

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  1. Nefton Pali
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Correspondence to Nefton Pali.

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The main result of this paper was also obtained by Fong (see [1]). Fong informed us that he was not aware of the results in [7, 8] when he wrote his preprint. One of the differences in our approach is that we use coordinate free notation, while Fong in [1] follows standard tradition of notation.

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Pali, N. Concavity of Perelman’s \(\mathcal {W}\)-functional over the space of Kähler potentials. European Journal of Mathematics 3, 587–602 (2017). https://doi.org/10.1007/s40879-017-0155-3

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  • DOI: https://doi.org/10.1007/s40879-017-0155-3

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