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On Representative Domains and Cartan’s Theorem

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Book cover Complex Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 144))

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Abstract

This is a short survey article on Cartan’s theorem about automorphisms fixing the origin for certain class of quasi-circular domains and non-hyperbolic circular domains. Some open problems are also given.

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References

  1. Boas, H.P.: Lu Qi-Keng’s problem. J. Korean Math. Soc. 37, 253–267 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Braun, R., Kaup, W., Upmeier, H.: On the automorphisms of circular and Reinhardt domains in complex Banach spaces. Manuscripta Math. 25, 97–133 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Harz, T., Shcherbina, N., Tomassini, G.: Wermer type sets and extension of CR functions. Indiana Univ. Math. J. 61, 431–459 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ishi, H., Kai, C.: The representative domain of a homogeneous bounded domain. Kyushu J. Math. 64, 35–47 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kaup, W.: Über das Randverhalten von holomorphen Automorphismen beschränkter Gebiete. Manuscripta Math. 3, 257–270 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kim, H., Ninh, V.T., Yamamori, A.: The automorphism group of a certain unbounded non-hyperbolic domain. J. Math. Anal. Appl. 409, 637–642 (2014)

    Article  MathSciNet  Google Scholar 

  7. Kim, H.: The automorphism group of an unbounded domain related to Wermer type sets. J. Math. Anal. Appl. 421, 1196–1206 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Krantz, S.G.: Function Theory of Several Complex Variables, 2nd edn. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, California (1992)

    MATH  Google Scholar 

  9. Lu, Q.-K.: On the representative domain, several complex variables (Hangzhou, 1981). Birkhäuser. Boston, pp. 199–211 (1984)

    Google Scholar 

  10. Narasimhan, R.: Several Complex Variables. Chicago lecture in mathematics. The University of Chicago Press, Chicago (1971)

    MATH  Google Scholar 

  11. Rong, F.: On automorphisms of quasi-circular domains fixing the origin. Bull. Sci. Math., (to appear)

    Google Scholar 

  12. Springer, G.: Pseudo-conformal transformations onto circular domains. Duke Math. J. 18, 411–424 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tsuboi, T.: Bergman representative domains and minimal domains. Japan. J. Math. 29, 141–148 (1959)

    MathSciNet  Google Scholar 

  14. Yamamori, A.: The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function. Complex Var. Elliptic Equ. 58(6), 783–793 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yamamori, A.: Automorphisms of normal quasi-circular domains. Bull. Sci. Math. 138, 406–415 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yamamori, A.: On the linearity of origin-preserving automorphisms of quasi-circular domains in \(\mathbb{C}^n\). J. Math. Anal. Appl. 426, 612–623 (2015)

    Article  MathSciNet  Google Scholar 

  17. Zapałowski, P.: Proper holomorphic mappings between symmetrized ellipsoids. Arch. Math. (Basel) 97(4), 373–384 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This article is based on the talk “On Bergman’s representative domains and origin-preserving automorphisms of quasi-circular domains” given by the author at the KSCV10 Symposium, August, 2014. The author would like to thank the organizers for their kind invitation and hospitality. The author also thanks the anonymous referee for valuable comments on this paper.

The research of the author is supported in part by SRC-GaiA (Center for Geometry and its Applications), the Grant 2011-0030044 from The Ministry of Education, The Republic of Korea.

Author information

Authors and Affiliations

  1. The Center for Geometry and Its Applications, Pohang University of Science and Technology, Pohang, 790-784, Republic of Korea

    Atsushi Yamamori

Authors
  1. Atsushi Yamamori
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Correspondence to Atsushi Yamamori .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Università di Roma Tor Vergata, Rome, Italy

    Filippo Bracci

  2. Dept. of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  3. Université Joseph Fourier Grenoble, Institut Fourier, Grenoble, France

    Hervé Gaussier

  4. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo, Japan

    Kengo Hirachi

  5. Department of Mathematics, POSTECH, Pohang, Korea (Republic of)

    Kang-Tae Kim

  6. Department of Mathematics, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Nikolay Shcherbina

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Yamamori, A. (2015). On Representative Domains and Cartan’s Theorem. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_26

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