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An extension of H. Cartan's theorem


Authors: So-chin Chen and Shih-Biau Jang
Journal: Proc. Amer. Math. Soc. 127 (1999), 2265-2271
MSC (1991): Primary 32M05
DOI: https://doi.org/10.1090/S0002-9939-99-04953-9
Published electronically: March 23, 1999
MathSciNet review: 1618717
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Abstract: In this article we prove that if $D\subset \mathbb{C}^n$, $n\ge 2$, is a bounded pseudoconvex domain with real analytic boundary, then for each $g(z)\in \mathrm{Aut}(D)$, there exists a fixed open neighborhood $\Omega _g$ of $\overline{D}$ and an open neighborhood $V_g$ of $g(z)$ in $\mathrm{Aut}(D)$ such that any $h(z)\in V_g$ can be extended holomorphically to $\Omega _g$, and that the action defined by

\begin{align*}\pi:& V_g\times \Omega _g\to \mathbb{C}^n\\ &(f,z)\mapsto \pi(f,z)=f(z) \end{align*}

is real analytic in joint variables. This extends H. Cartan's theorem beyond the boundary. Some applications are also discussed here.


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Additional Information

So-chin Chen
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China

Shih-Biau Jang
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China

DOI: https://doi.org/10.1090/S0002-9939-99-04953-9
Keywords: Automorphism group, pseudoconvex domains, condition $R$
Received by editor(s): October 21, 1997
Published electronically: March 23, 1999
Additional Notes: Both authors are partially supported by a grant NSC 85-2121-M-007-028 from the National Science Council of the Republic of China.
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society

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