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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Geometry of quasi-circular domains and applications to tetrablock


Author: Łukasz Kosiński
Journal: Proc. Amer. Math. Soc. 139 (2011), 559-569
MSC (2010): Primary 32H35, 32A07
Published electronically: July 16, 2010
MathSciNet review: 2736338
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Abstract: We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains.

Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock).

It is also shown that the tetrablock is not $ \mathbb{C}$-convex.


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Additional Information

Łukasz Kosiński
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Łojasiewicza 6, 30-348 Kraków, Poland
Email: lukasz.kosinski@im.uj.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10493-8
PII: S 0002-9939(2010)10493-8
Keywords: Tetrablock, proper holomorphic mappings, group of automorphisms, quasi-circular domains, Shilov boundary.
Received by editor(s): November 11, 2009
Received by editor(s) in revised form: November 12, 2009, and March 10, 2010
Published electronically: July 16, 2010
Additional Notes: This work was partially supported by the Research Grant of the Polish Ministry of Science and Higher Education N$^{o}$ N N201 271435.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.