Complex Analysis
On the stability group of CR manifolds
Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 169-172.

For any essentially finite minimal real-analytic generic submanifold MCN, N2, we show that for every point pM the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their =(p) jets at p. As an application, we derive a Lie group structure for the stability group Aut(M,p). We also show that the order =(p) of the jet space in which the group Aut(M,p) embeds can be chosen to depend upper-semicontinuously on p. This yields that given any compact real-analytic minimal CR submanifold M in CN, there exists an integer k depending only on M such that for every point pM local CR diffeomorphisms mapping a neighbourhood of p in M into another real-analytic CR submanifold in CN with the same CR dimension as that of M are uniquely determined by their k-jet at p.

Pour toute sous-variété analytique réelle générique essentiellement finie et minimale MCN, N2, nous montrons que pour tout point pM, les automorphismes CR locaux analytiques réels de M fixant p sont paramétrés analytiquement par leur =(p)-jets en p. Comme application, nous obtenons une structure de groupe de Lie sur le groupe d'isotropie Aut(M,p). Nous montrons aussi que l'ordre =(p) de l'espace des jets dans lequel le groupe Aut(M,p) se plonge peut être choisi de façon à ce que l'application p(p) soit semi-continue supérieurement. En corollaire, nous obtenons qu'étant donnée toute sous-variété CR compacte analytique réelle et minimale MCN, il existe un entier positif k, dépendant uniquement de M, tel que pour tout point pM les difféomorphismes CR locaux envoyant un voisinage de p dans M sur toute autre sous-variété CR de CN de même dimension CR que celle de M sont uniquement déterminés par leur k-jet en p.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2006.06.016
Lamel, Bernhard 1; Mir, Nordine 2

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 Étienne du Rouvray, France
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Lamel, Bernhard; Mir, Nordine. On the stability group of CR manifolds. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 169-172. doi : 10.1016/j.crma.2006.06.016. http://www.numdam.org/articles/10.1016/j.crma.2006.06.016/

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