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On the Automorphism Groups of Models in \(\mathbb {C}^{2}\)

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Abstract

In this article, we first give a characterization of models in \(\mathbb {C}^{2}\) by their noncompact automorphism groups. Then we give an explicit description for automorphism groups of models in \(\mathbb {C}^{2}\).

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References

  1. Ahn, H., Byun, J., Park, J.D.: Automorphisms of the Hartogs type domains over classical symmetric domains. Int. J. Math. 23(9), 1250098 (11 pages) (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bedford, E., Pinchuk, S.: Domains in \(\mathbb {C}^{n+1}\) with noncompact automorphism group. J. Geom. Anal. 1, 165–191 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bedford, E., Pinchuk, S.: Domains in \(\mathbb {C}^{2}\) with noncompact automorphism groups. Indiana Univ. Math. J. 47, 199–222 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bell, S.: Local regularity of C.R. homeomorphisms. Duke Math. J. 57, 295–300 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bell, S., Catlin, D.: Regularity of CR mappings. Math. Z. 199(3), 357–368 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berteloot, F.: Attraction de disques analytiques et continuité Holdérienne d’applications holomorphes propres. Topics in Compl. Anal., Banach Center Publ., pp. 91–98 (1995)

  7. Berteloot, F.: Characterization of models in \(\mathbb {C}^{2}\) by their automorphism groups. Int. J. Math. 5, 619–634 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byun, J., Cho, H.R.: Explicit description for the automorphism group of the Kohn-Nirenberg domain. Math. Z. 263(2), 295–305 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byun, J., Cho, H.R.: Explicit description for the automorphism group of the Fornaess domain. J. Math. Anal. Appl. 369(1), 10–14 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, S.-C.: Characterization of automorphisms on the Barrett and the Diederich-Fornaess worm domains. Trans. Am. Math. Soc. 338(1), 431–440 (1993)

    MathSciNet  MATH  Google Scholar 

  11. D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115, 615–637 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Diederich, K., Pinchuk, S.: Proper holomorphic maps in dimension 2 extend. Indiana Univ. Math. J. 44(4), 1089–1126 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Do, D.T., Ninh, V.T.: Characterization of domains in \(\mathbb {C}^{n}\) by their noncompact automorphism groups. Nagoya Math. J. 196, 135–160 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Duren, P.: Univalent functions. Grundlehren der Mathematischen Wissenschaften 259. Springer (1983)

  15. Greene, R., Krantz, S.G.: Biholomorphic self-maps of domains. Lect. Notes Math. 1276, 136–207 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Greene, R., Kim, K.-T., Krantz, S.G.: The Geometry of Complex Domains. Prog. Math., 291. Birkhuser Boston, Inc., Boston (2011)

    Book  Google Scholar 

  17. Fu, S., Wong, B.: On boundary accumulation points of a smoothly bounded pseudoconvex domain in \(\mathbb {C}^{2}\). Math. Ann. 310, 183–196 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Isaev, A., Krantz, S.G.: On the boundary orbit accumulation set for a domain with noncompact automorphism group. Michigan Math. J. 43, 611–617 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Isaev, A., Krantz, S.G.: Domains with non-compact automorphism group: A survey. Adv. Math. 146, 1–38 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jarnicki, M., Pflug, P.: On automorphisms of the symmetrized bidisc. Arch. Math. (Basel) 83(3), 264–266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kim, K.-T.: Automorphism groups of certain domains in \(\mathbb {C}^{n}\) with a singular boundary. Pacific J. Math. 151(1), 57–64 (1991)

  22. Kolar, M.: Normal forms for hypersurfaces of finite type in \(\mathbb {C}^{2}\). Math. Res. Lett. 12, 897–910 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Krantz, S.G.: The automorphism group of a domain with an exponentially flat boundary point. J. Math. Anal. Appl. 385(2), 823–827 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Oeljeklaus, K.: On the automorphism group of certain hyperbolic domains in \(\mathbb {C}^{2}\). Colloque d’Analyse Complexe et Gomtrie (Marseille, 1992). Astérisque 217(7), 193–216 (1993)

    MathSciNet  Google Scholar 

  25. Rosay, J.P.: Sur une caracterisation de la boule parmi les domaines de \(\mathbb {C}^{n}\) par son groupe d’automorphismes. Ann. Inst. Fourier 29(4), 91–97 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shafikov, R., Verma, K.: A local extension theorem for proper holomorphic mappings in \(\mathbb {C}^{2}\). J. Geom. Anal. 13(4), 697–714 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shimizu, S.: Automorphisms of bounded Reinhardt domains. Proc. Japan Acad. Ser. A Math. Sci. 63(9), 354–355 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sunada, T.: Holomorphic equivalence problem for bounded Reinhardt domains. Math. Ann. 235(2), 111–128 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Verma, K.: A characterization of domains in \(\mathbb {C}^{2}\) with noncompact automorphism group. Math. Ann. 334(3–4), 645–701 (2009)

  30. Wong, B.: Characterization of the ball in \(\mathbb {C}^{n}\) by its automorphism group. Invent. Math. 41, 253–257 (1977)

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Acknowledgments

We would like to thank Prof. Kang-Tae Kim, Prof. Do Duc Thai, and Dr. Hyeseon Kim for their precious discussions on this material. Especially, we would like to express our gratitude to the referees for many helpful comments.

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Author notes

  1. Ninh Van Thu

    Present address: Department of Mathematics, Vietnam National University at Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

Authors and Affiliations

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

    Ninh Van Thu

  2. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy str., Hanoi, Vietnam

    Mai Anh Duc

Authors
  1. Ninh Van Thu
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  2. Mai Anh Duc
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Correspondence to Ninh Van Thu.

Additional information

The research of the first author was supported in part by an NRF grant 2011-0030044 (SRC-GAIA) of the Ministry of Education, The Republic of Korea. The research of the authors was supported in part by an NAFOSTED grant of Vietnam.

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Van Thu, N., Duc, M.A. On the Automorphism Groups of Models in \(\mathbb {C}^{2}\) . Acta Math Vietnam 41, 457–470 (2016). https://doi.org/10.1007/s40306-015-0160-x

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  • DOI: https://doi.org/10.1007/s40306-015-0160-x

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