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Proper Holomorphic Mappings among Bounded Symmetric Domains and Related Hartogs Domains

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The classical Alexander’s Theorem states that every proper holomorphic self-mapping of a complex unit ball of dimension at least 2 is an automorphism. Since then, the study of proper holomorphic mappings has become an important topic in several complex variables. Bounded symmetric domains, which include the complex unit balls, are among the most important domains in complex Euclidean spaces, due to the fact that they possess a lot of symmetries and are the universal covering spaces of various important mathematical objects. Henkin and Novikov proved that the analogue of Alexander’s Theorem is also true for irreducible bounded symmetric domains of higher rank. These rigidity results for proper holomorphic mappings among bounded symmetric domains have been, by the efforts of a lot of people, extended to the cases with positive co-dimension or rank difference. The purpose of this article is to give a survey for these developments. In addition, we also include a section discussing some generalizations to the Hartogs domains over irreducible bounded symmetric domains.

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Acknowledgements

We thank the referees for very helpful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, P. R. China

    Sui-chung Ng

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China

    Zhenhan Tu & Wanke Yin

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  1. Sui-chung Ng
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  2. Zhenhan Tu
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  3. Wanke Yin
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Correspondence to Zhenhan Tu.

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Dedicated to Professor Zhihua Chen’s memory

S. C. Ng is supported in part by National Natural Science Foundation of China (Grant No. 12471078) and Science and Technology Commission of Shanghai Municipality (Grant No. 22DZ2229014). Z. Tu is supported by the National Natural Science Foundation of China (Grant Nos. 12571089 and 12361131577). W. Yin is supported by the National Natural Science Foundation of China (Grant Nos. 12171372 and 12361131577). Z. Tu and W. Yin are supported by Hubei Provincial Innovation Group Project (Grant No. 2025AFA044)

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Ng, Sc., Tu, Z. & Yin, W. Proper Holomorphic Mappings among Bounded Symmetric Domains and Related Hartogs Domains. Acta. Math. Sin.-English Ser. (2025). https://doi.org/10.1007/s10114-025-3377-1

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