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On the localization principle for the automorphisms of pseudoellipsoids

Authors: Mario Landucci and Andrea Spiro
Journal: Proc. Amer. Math. Soc. 137 (2009), 1339-1345
MSC (2000): Primary 32H12, 32H02, 32H35
Published electronically: December 3, 2008
MathSciNet review: 2465657
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Abstract: We show that Alexander's extendibility theorem for a local automorphism of the unit ball is valid also for a local automorphism $ f$ of a pseudoellipsoid $ \mathcal{E}^n_{(p_1, \dots, p_{k})}\overset{\text{def}}{=} \{ z \in \mathbb{C}... ...\vert^2 + \vert z_{n-k+1}\vert^{2 p_1} + \dots + \vert z_n\vert^{2 p_{k}} < 1\}$, provided that $ f$ is defined on a region $ \mathcal{U} \subset \mathcal{E}^n_{(p)}$ such that: i) $ \partial \mathcal{U} \cap \partial \mathcal{E}^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $ \mathcal{U}\cap\{ z_i = 0 \} \neq \emptyset$ for any $ n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.

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  • [Al] H. Alexander, Holomorphic mappings from the ball and polydisc, Math. Ann. 209 (1974), 249-256. MR 0352531 (50:5018)
  • [DS] G. Dini and A. Selvaggi Primicerio, Localization principle of automorphisms on generalized pseudoellipsoids, J. Geom. Anal. 7 (4) (1997), 575-584. MR 1669231 (99m:32032)
  • [FR] F. Forstnerič and J.-P. Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987), 239-252. MR 919504 (89c:32070)
  • [KLS] K.-T. Kim, M. Landucci and A. Spiro, Factorization of proper holomorphic mappings through Thullen domains, Pacific J. Math. 189 (2) (1999), 293-310. MR 1696125 (2000e:32021)
  • [KS] K.-T. Kim and A. Spiro, Moduli space of ramified holomorphic coverings of $ B^2$, in ``Complex geometric analysis in Pohang (1997)'', pp. 227-239, Contemp. Math., 222, Amer. Math. Soc., Providence, RI, 1999. MR 1653055 (99m:32033)
  • [La] M. Landucci, On the proper holomorphic equivalence for a class of pseudoconvex domains, Trans. Amer. Math. Soc. 282 (2) (1984), 807-811. MR 732122 (85a:32033)
  • [LS] M. Landucci and A. Spiro, Proper holomorphic maps between complete Reinhardt domains in $ C^2$, Complex Var. Theory Appl. 29 (1) (1996), 9-25. MR 1381999 (97a:32028)
  • [Pi] S. Pinčuk, The analytic continuation of holomorphic mappings, Math. USSR Sb. 27 (1975) 375-392 (translation from Mat. Sb. (N.S.) 98 (140) (1975), 416-435). MR 0393562 (52:14371)
  • [Pi1] S. Pinčuk, Holomorphic mappings of real-analytic hypersurfaces, Math. USSR Sb. 34 (1978), 503-519. MR 496595 (80c:32022)
  • [Ru] W. Rudin, Holomorphic maps that extend to automorphisms of a ball, Proc. Amer. Math. Soc. 81 (1981), 429-432. MR 597656 (82c:32012)
  • [Ve] E. Vesentini, Capitoli scelti della teoria delle funzioni olomorfe, Unione Matematica Italiana, Bologna: Pitagora Ed., 1984.
  • [We] S. Webster, Biholomorphic mappings and the Bergman kernel off the diagonal, Invent. Math. 51 (1979), 155-169. MR 528021 (81e:32029)

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Additional Information

Mario Landucci
Affiliation: Dip. Matematica Applicata “G. Sansone”, Università di Firenze, Via di Santa Marta 3, I-50139 Firenze, Italy

Andrea Spiro
Affiliation: Dip. Matematica e Informatica, Università di Camerino, Via Madonna delle Carceri, I-62032 Camerino (Macerata), Italy

Keywords: Alexander theorem, pseudoellipsoids, localization principle
Received by editor(s): June 25, 2007
Received by editor(s) in revised form: February 17, 2008
Published electronically: December 3, 2008
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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