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Three circle theorem on almost Hermitian manifolds and applications

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Abstract

In this paper, we extend Gang Liu’s three circle theorem for Kähler manifolds to almost Hermitian manifolds. As applications of the three circle theorem, we obtain sharp dimension estimates for spaces of holomorphic functions of polynomial growth and rigidity for the estimates, Liouville theorems for pluri-subharmonic functions of sub-logarithmic growth and Liouville theorems for holomorphic functions of Cheng-type, and fundamental theorem of algebra on almost Hermitian manifolds with slightly negative holomorphic sectional curvature. The results are generalization of Liu’s results and some of them are new even for the Kähler case. We also discuss the converse of the three circle theorem on Hermitian manifolds which turns out to be rather different with the Kähler case. In order to obtain the three circle theorem on almost Hermitian manifolds, we also establish a general maximum principle in the spirit of Calabi’s trick so that a general three circle theorem is a straightforward consequence of the general maximum principle. The general maximum principle and three circle theorem established may be of independent interests.

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Acknowledgements

The authors would like to thank Professor Jeffrey D. Streets for informing them the important works [23, 28, 29] on Hermitian curvature flow and X.-K. Yang’s work [30] on Chern-Ricci flow.

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China

    Chengjie Yu & Chuangyuan Zhang

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  1. Chengjie Yu
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  2. Chuangyuan Zhang
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Correspondence to Chengjie Yu.

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Communicated by S. A. Chang.

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C. Yu: Research partially supported by GDNSF with contract no. 2021A1515010264 and NNSF of China with contract no. 11571215.

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Yu, C., Zhang, C. Three circle theorem on almost Hermitian manifolds and applications. Calc. Var. 61, 184 (2022). https://doi.org/10.1007/s00526-022-02294-5

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