Parametrization of local CR automorphisms by finite jets and applications

Lamel, Bernhard¹ ; Mir, Nordine²

¹ Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Austria
² Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Avenue de l’Université, B.P. 12, 76801 Saint Etienne du Rouvray, France

Journal of the American Mathematical Society, Tome 20 (2007) no. 2, pp. 519-572

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-06-00534-0

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