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Some Recent Results on Holomorphic Isometries of the Complex Unit Ball into Bounded Symmetric Domains and Related Problems

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Geometric Complex Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 246))

Abstract

In his seminal work Calabi established the foundation on the study of holomorphic isometries from a Kähler manifold with real analytic local potential functions into complex space forms, e.g., Fubini-Study spaces. This leads to interior extension results on germs of holomorphic isometries between bounded domains. General results on boundary extension were obtained by Mok under assumptions such as the rationality of Bergman kernels, which applies especially to holomorphic isometries between bounded symmetric domains in their Harish-Chandra realizations. Because of rigidity results in the cases where the holomorphic isometry is defined on an irreducible bounded symmetric domain of rank \({\ge }2\), we focus on holomorphic isometries defined on the complex unit ball \(\mathbb B^n, n \ge 1\). We discuss results on the construction, characterization and classification of holomorphic isometries of the complex unit ball into bounded symmetric domains and more generally into bounded homogeneous domains. Furthermore, in relation to the study of the Hyperbolic Ax-Lindemann Conjecture for not necessarily arithmetic quotients of bounded symmetric domains, such holomorphic isometric embeddings play an important role. We also present some differential-geometric techniques arising from the study of the latter conjecture.

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Acknowledgements

This research was partially supported by the GRF 17303814, of the HKRGC, Hong Kong. The author would like to thank the organizers of KSCV for their invitation, and it is a pleasure for him to dedicate this article to Professor Kang-Tae Kim, initiator of this long series of international symposia in several complex variables, on the occasion of his sixtieth birthday.

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Authors and Affiliations

  1. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

    Ngaiming Mok

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  1. Ngaiming Mok
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Correspondence to Ngaiming Mok .

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Editors and Affiliations

  1. Department of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  2. Department of Mathematics, Pusan National University, Busan, Korea (Republic of)

    Hong Rae Cho

  3. Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Korea (Republic of)

    Sung Yeon Kim

  4. Department of Mathematics, Gyeongsang National University, Jinju, Korea (Republic of)

    Kang-Hyurk Lee

  5. Department of Mathematics, Kyung Hee University, Seoul, Korea (Republic of)

    Jong-Do Park

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Mok, N. (2018). Some Recent Results on Holomorphic Isometries of the Complex Unit Ball into Bounded Symmetric Domains and Related Problems. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_21

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