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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Parametrization of local CR automorphisms by finite jets and applications
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by Bernhard Lamel and Nordine Mir PDF
J. Amer. Math. Soc. 20 (2007), 519-572 Request permission

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 $H=\textrm {Id}$. 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
  • MR Author ID: 685199
  • ORCID: 0000-0002-6322-6360
  • 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
  • Received by editor(s): June 10, 2005
  • Published electronically: April 25, 2006
  • Additional Notes: The first author was supported by the FWF, Projekt P17111.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2276779