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Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

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Geometric Complex Analysis

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

Abstract

In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most of them pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map \(X\rightarrow Y\) between Stein manifolds is homotopic to a proper holomorphic embedding provided that \(\dim Y>2\dim X\) and we allow a homotopic deformation of the Stein structure on X.

Dedicated to Kang-Tae Kim for his sixtieth birthday.

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Acknowledgements

The author is supported in part by the research program P1-0291 and grants J1-7256 and J1-9104 from ARRS, Republic of Slovenia. I wish to thank Antonio Alarcón and Rafael Andrist for a helpful discussion concerning Corollary 3.5 and the Schoen-Yau conjecture, Barbara Drinovec Drnovšek for her remarks on the exposition, Josip Globevnik for the reference to the paper of Božin [25], Frank Kutzschebauch for having proposed to include the material in Sect. 2.4, and Peter Landweber for his remarks which helped me to improve the language and presentation.

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  1. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, SI, Slovenia

    Franc Forstnerič

  2. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, SI, Slovenia

    Franc Forstnerič

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  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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Forstnerič, F. (2018). Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey. 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_11

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