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Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold

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Abstract

In this paper, we study the uniformly strong convergence of the Kähler-Ricci flow on a Fano manifold with varied initial metrics and smoothly deformed complex structures. As an application, we prove the uniqueness of Kähler-Ricci solitons in the sense of diffeomorphism orbits. The result generalizes Tian-Zhu’s theorem for the uniqueness of of Kähler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun’s result of the uniqueness of Kähler-Einstein metric orbits.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11971423) and the Fundamental Research Funds for the Central Universities. The second author was supported by National Natural Science Foundation of China (Grant No. 11771019), Beijing Science Foundation (Grant No. Z180004) and National Key R&D Program of China (Grant No. SQ2020YFA070059). The authors thank Professor Gang Tian for inspiring conversations. The authors also thank Chi Li for telling them that the uniqueness of algebraic structures of \({\tilde M_\infty}\) in Theorem 1.1 was solved by using non-Archimedean geometry in his recent joint paper with Han [23].

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Feng Wang

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Xiaohua Zhu

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  1. Feng Wang
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  2. Xiaohua Zhu
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Correspondence to Feng Wang.

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Wang, F., Zhu, X. Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold. Sci. China Math. 65, 2337–2370 (2022). https://doi.org/10.1007/s11425-021-1928-1

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