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On a twisted conical Kähler–Ricci flow

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Abstract

In this paper, we discuss diameter bound and Gromov–Hausdorff convergence of a twisted conical Kähler–Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical Kähler metric with possibly unbounded scalar curvature, the conical Kähler–Ricci flow will instantly have bounded scalar curvature for \(t>0\), and the bound is of the form \(\frac{C}{t}\). Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted Kähler–Ricci flow and its convergence.

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Notes

  1. We point out that in previous works on bounding scalar curvature along Kähler–Ricci flow (see [6, 23, 31, 49]), people usually choose \(s=1\) to define this quantity E. Here we choose \(s<1\) to slightly simplify some arguments.

  2. Note that choosing the positive number s to be in (0, 1) helps to make sure the factor \(s-s^2\) appeared in (4.25) is positive and avoid an additional trick used in [6, 31].

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Acknowledgements

The author thanks Prof. Huai-Dong Cao, Prof. Gang Tian, Prof. Zhenlei Zhang and Dr. Shaochuang Huang for useful discussions, Dr. Jiawei Liu for valuable comments and the referee for careful reading and valuable suggestions and comments. Part of this work was carried out while the author was visiting University of Macau and Capital Normal University, which he would like to thank for the hospitality.

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  1. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Yashan Zhang

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Zhang, Y. On a twisted conical Kähler–Ricci flow. Ann Glob Anal Geom 55, 69–98 (2019). https://doi.org/10.1007/s10455-018-9619-z

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