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Proper holomorphic mappings of balanced domains in \(\mathbb {C}^n\)


We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in \(\mathbb {C}^n, \ n > 1\). Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in \(\mathbb {C}^n, \ n > 1\), is an automorphism. The main novelty of our proof is the use of a recent result of Opshtein on the behaviour of the iterates of holomorphic self-maps of a certain class of domains. We use Opshtein’s theorem, together with the tools made available by finiteness of type, to deduce that the aforementioned map is unbranched. The monodromy theorem then delivers the result.

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I would like to thank my advisor Gautam Bharali for his support during the course of this work, and for suggesting several useful ideas. The idea behind Lemma 5.1 was given by him. I would also like to thank my colleague and friend G. P. Balakumar for some informative discussions related to this work. I am truly grateful to the anonymous referee of this article for the many suggestions for improving the exposition.

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Correspondence to Jaikrishnan Janardhanan.

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This work is supported by a UGC Centre for Advanced Study Grant and by a scholarship from the IISc.

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Janardhanan, J. Proper holomorphic mappings of balanced domains in \(\mathbb {C}^n\) . Math. Z. 280, 257–268 (2015).

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  • Balanced domains
  • Proper holomorphic maps
  • Alexander’s theorem

Mathematics Subject Classification

  • Primary 32H35