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An Almost Complex Chern–Ricci Flow

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We consider the evolution of an almost Hermitian metric by the (1, 1) part of its Chern–Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern–Ricci flow if the complex structure is integrable and with the Kähler–Ricci flow if moreover the initial metric is Kähler. We find the maximal existence time for the flow in term of the initial data and also give some convergence results. As an example, we study this flow on the (locally) homogeneous manifolds in more detail.

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Acknowledgements

The author thanks Professor Valentino Tosatti for suggesting him this question and for many other invaluable conversations and directions. The author also thanks Professor Jean-Pierre Demailly and Professor Ben Weinkove for some useful comments. The author is also grateful to the anonymous referees and the editor for their careful reading and helpful suggestions which greatly improved the paper. This study is supported by the National Natural Science Foundation of China Grant No. 11401023, and the author’s post-doc is supported by the European Research Council (ERC) Grant No. 670846 (ALKAGE).

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  1. Institut Fourier, Université Grenoble Alpes, 100 rue des maths, 38610, Gières, France

    Tao Zheng

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  1. Tao Zheng
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Zheng, T. An Almost Complex Chern–Ricci Flow. J Geom Anal 28, 2129–2165 (2018). https://doi.org/10.1007/s12220-017-9898-9

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