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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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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Additional Information

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

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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09726-8
PII: S 0002-9939(08)09726-8
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.