Geometry of quasi-circular domains and applications to tetrablock

Kosiński, Łukasz

doi:10.1090/S0002-9939-2010-10493-8

Geometry of quasi-circular domains and applications to tetrablock
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Proc. Amer. Math. Soc. 139 (2011), 559-569 Request permission

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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