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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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  • [Ab-Wh-Yo] A. Abouhajar, M. White, and N. Young, A Schwarz lemma for a domain related to $ \mu$-synthesis, Journal of Geometric Analysis 17 (2007), no. 4, 717-750. MR 2365665 (2009e:32008)
  • [Bel1] S. Bell, Analytic hypoellipticity of the $ \bar \partial $-Neumann problem and extendability of holomorphic mappings, Acta Math. 147 (1981), no. 1-2, 109-116. MR 631091 (83a:32011)
  • [Bel2] S. Bell, Proper holomorphic mappings between circular domains, Comment. Math. Helv. 57 (1982), no. 4, 532-538. MR 694605 (84m:32032)
  • [Edi-Zwo1] A. Edigarian and W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. 84 (2005), 364-374. MR 2135687 (2006b:32020)
  • [Edi-Zwo2] A. Edigarian and W. Zwonek, Schwarz lemma for the tetrablock, Bulletin of the London Mathematical Society 41 (2009), no. 3, 506-514. MR 2506834 (2010e:32007)
  • [Har-Wri] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publ., 1978. MR 0067125 (16:673c)
  • [Hen-Nov] G. M. Henkin and R. Novikov, Proper mappings of classical domains, Lecture Notes in Math. 1043, Springer-Verlag, 1984, pp. 625-627.
  • [Hua] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., Providence, RI, 1963. MR 0171936 (30:2162)
  • [Jac] D. Jacquet, $ \mathbb{C}$-convex domains with $ C\sp 2$ boundary, Complex Var. Elliptic Equ. 51 (2006), no. 4, 303-312. MR 2218722 (2007j:32005)
  • [Lem] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. MR 660145 (84d:32036)
  • [Mok] N. Mok, Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains, Chinese Annals of Mathematics - Series B 29 (2008), no. 2, 135-146. MR 2392329 (2009i:32024)
  • [Nar] R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, 1971. MR 0342725 (49:7470)
  • [Nik] N. Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann. Polon. Math. 88 (2006), 279-283. MR 2260407 (2007f:32024)
  • [Rud1] W. Rudin, Function theory in the unit ball of $ \mathbb{C}^n$, Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, 1980. MR 601594 (82i:32002)
  • [Rud2] W. Rudin, Holomorphic maps that extend to automorphisms of a ball, Proc. Amer. Math. Soc. 81 (1981), no. 3, 429-432. MR 597656 (82c:32012)
  • [Tu1] Z.-H. Tu, Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains, Proc. Amer. Math. Soc. 130 (2002), 1035-1042. MR 1873777 (2003a:32027)
  • [Tu2] Z.-H. Tu, Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains, Math. Z. 240 (2002), 13-35. MR 1906705 (2003g:32034)
  • [Tu3] Z.-H. Tu, Rigidity of proper holomorphic mappings between bounded symmetric domains, Geometric function theory in several complex variables, pp. 310-316, World Sci. Publishing, River Edge, NJ, 2004. MR 2115800 (2005m:32032)
  • [Tum-Hen] A. E. Tumanov and G. M. Khenkin, Local characterization of holomorphic automorphisms of Siegel domains, Funktsional. Anal. i Prilozhen. 170 (1983), no. 4, 49-61 (Russian). MR 725415 (86a:32063)
  • [You] N. Young, The automorphism group of the tetrablock, Journal of the London Mathematical Society 77 (2008), no. 3, 757-770. MR 2418303 (2009a:32032)

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Additional Information

Łukasz Kosiński
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Łojasiewicza 6, 30-348 Kraków, Poland

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.

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