• DOI:10.1007/s00209-015-1421-z
• Corpus ID: 119147286

Proper holomorphic mappings of balanced domains in $$\mathbb {C}^n$$Cn

@article{Janardhanan2014ProperHM,
  title={Proper holomorphic mappings of balanced domains in \$\$\mathbb \{C\}^n\$\$Cn},
  author={Jaikrishnan Janardhanan},
  journal={Mathematische Zeitschrift},
  year={2014},
  volume={280},
  pages={257-268},
  url={https://api.semanticscholar.org/CorpusID:119147286}
}

• Jaikrishnan Janardhanan
• Published 18 April 2014
• Mathematics
• Mathematische Zeitschrift

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in $$\mathbb {C}^n, \ n > 1$$Cn,n>1. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in $$\mathbb {C}^n, \ n > 1$$Cn,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… 

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in $$\mathbb {C}^n, \ n > 1$$Cn,n>1. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in $$\mathbb {C}^n, \ n > 1$$Cn,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.