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Parametrization of local CR automorphisms by finite jets and applications


Authors: Bernhard Lamel and Nordine Mir
Journal: J. Amer. Math. Soc. 20 (2007), 519-572
MSC (2000): Primary 32H02, 32H12, 32V05, 32V15, 32V20, 32V25, 32V35, 32V40
DOI: https://doi.org/10.1090/S0894-0347-06-00534-0
Published electronically: April 25, 2006
MathSciNet review: 2276779
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Abstract: For any real-analytic hypersurface $ M\subset \mathbb{C}^N$, which does not contain any complex-analytic subvariety of positive dimension, we show that for every point $ p\in M$ the local real-analytic CR automorphisms of $ M$ fixing $ p$ can be parametrized real-analytically by their $ \ell_p$ jets at $ p$. As a direct application, we derive a Lie group structure for the topological group $ \operatorname{Aut}(M,p)$. Furthermore, we also show that the order $ \ell_p$ of the jet space in which the group $ \operatorname{Aut}(M,p)$ embeds can be chosen to depend upper-semicontinuously on $ p$. As a first consequence, it follows that given any compact real-analytic hypersurface $ M$ in $ \mathbb{C}^N$, there exists an integer $ k$ depending only on $ M$ such that for every point $ p\in M$ germs at $ p$ of CR diffeomorphisms mapping $ M$ into another real-analytic hypersurface in $ \mathbb{C}^N$ are uniquely determined by their $ k$-jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain $ \Omega$ with smooth real-analytic boundary, there exists an integer $ k$ depending only on $ \partial \Omega$ such that if $ H\colon \Omega\to \Omega$ is a proper holomorphic mapping extending smoothly up to $ \partial \Omega$ near some point $ p\in \partial \Omega$ with the same $ k$-jet at $ p$ with that of the identity mapping, then necessarily $ H={\rm Id}$.

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.


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Additional Information

Bernhard Lamel
Affiliation: Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Austria
Email: lamelb@member.ams.org

Nordine Mir
Affiliation: Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Avenue de l’Université, B.P. 12, 76801 Saint Etienne du Rouvray, France
Email: Nordine.Mir@univ-rouen.fr

DOI: https://doi.org/10.1090/S0894-0347-06-00534-0
Keywords: CR automorphism, jet parametrization, finite jet determination, singular analytic equations
Received by editor(s): June 10, 2005
Published electronically: April 25, 2006
Additional Notes: The first author was supported by the FWF, Projekt P17111.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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