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On convergence criteria for the coupled flow of Li–Yuan–Zhang

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Abstract

A one-parameter family of coupled flows depending on a parameter \(\kappa >0\) is introduced which reduces when \(\kappa =1\) to the coupled flow of a metric \(\omega \) with a (1, 1)-form \(\alpha \) due recently to Y. Li, Y. Yuan, and Y. Zhang. It is shown in particular that, for \(\kappa \not =1\), estimates for derivatives of all orders would follow from \(C^0\) estimates for \(\omega \) and \(\alpha \). Together with the monotonicity of suitably adapted energy functionals, this can be applied to establish the convergence of the flow in some situations, including on Riemann surfaces. Very little is known as yet about the monotonicity and convergence of flows in presence of couplings, and conditions such as \(\kappa \not =1\) seem new and may be useful in the future.

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Notes

  1. A similar decomposition was used by Trudinger and Wang [24] for the affine Plateau problem.

  2. In keeping with the traditional notation for the Mabuchi functional itself, we have denoted the volume \(\int _X \omega _0^n\) of \(\omega \) by V instead of normalizing it to 1 as in the previous sections. This will also allow normalizing \(\lambda \) to \(-1\) in subsequent sections.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Teng Fei, Bin Guo & Duong H. Phong

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  1. Teng Fei
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  2. Bin Guo
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  3. Duong H. Phong
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Correspondence to Duong H. Phong.

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Work supported in part by the National Science Foundation under Grants DMS-12-66033 and DMS-17-10500.

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Fei, T., Guo, B. & Phong, D.H. On convergence criteria for the coupled flow of Li–Yuan–Zhang. Math. Z. 292, 473–497 (2019). https://doi.org/10.1007/s00209-019-02272-2

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