On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

[Sur l'enveloppe d'holomorphie de domaines recouverts par des "chapeaux" Levi-plats et le principe de réflexion]

Merker, Joël

Annales de l'Institut Fourier, Tome 52 (2002), p. 1443-1523 / Harvested from Numdam

Résumé

Dans cet article, nous associons les techniques du principe de réflexion de Lewy-Pinchuk avec celles du principe de continuité de Behnke-Sommer. Après avoir prolongé holomorphiquement une fonction dite “de réflexion” à une congruence de sous-variétés de Segre, nous sommes conduits à l’étude de l’enveloppe d’holomorphie d’un domaine recouvert d’un “chapeau” Levi-plat lisse. D’après notre résultat principal, tout CR-difféomorphisme $h: M\rightarrow M^{\prime }$ de classe ${\mathcal {C}}^\infty $ entre deux hypersurfaces analytiques réelles globalement minimales de ${\mathbb {C}}^n$ ($n\ge 2$) est analytique réel en tout point de $M$ si $M^{\prime }$ est holomorphiquement non-dégénérée. Plus généralement, nous établissons que la fonction de réflexion ${\mathcal {R}}_h^{\prime }$ associée à un tel difféomorphisme CR de classe ${\mathcal {C}}^\infty $ entre deux hypersurfaces analytiques réelles globalement minimales se prolonge toujours holomorphiquement à un voisinage du graphe de $\bar{h}$ dans $M\times \overline{M}^{\prime }$, sans aucune condition de non-dégénérescence sur $M^{\prime }$. Cet énoncé fournit une nouvelle version du principe de réflexion de Schwarz en plusieurs variables complexes. Enfin, nous démontrons que toute application $h:M\rightarrow M^{\prime }$ de classe ${\mathcal {C}}^\infty $ et CR entre deux hypersurfaces analytiques réelles ne contenant pas de courbes holomorphes est analytique réelle en tout point de $M$, sans aucune condition de rang sur $h$.

Abstract

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat “hat”. In our main theorem, we show that every ${\mathcal {C}}^\infty $-smooth CR diffeomorphism $h: M\rightarrow M^{\prime }$ between two globally minimal real analytic hypersurfaces in ${\mathbb {C}}^n$ ($n\ge 2$) is real analytic at every point of $M$ if $M^{\prime }$ is holomorphically nondegenerate. More generally, we establish that the reflection function ${\mathcal {R}}_h^{\prime }$ associated to such a ${\mathcal {C}}^\infty $-smooth CR diffeomorphism between two globally minimal hypersurfaces in ${\mathbb {C}}^n$ ($n\ge 1$) always extends holomorphically to a neighborhood of the graph of $\bar{h}$ in $M\times \overline{M}^{\prime }$, without any nondegeneracy condition on $M^{\prime }$. This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every ${\mathcal {C}}^\infty $- smooth CR mapping $h: M\rightarrow M^{\prime }$ between two real analytic hypersurfaces containing no complex curves is real analytic at every point of $M$, without any rank condition on $h$.

Détail

Publié le : 2002-01-01

MR 1935554 | Zbl 1031.32028

DOI : https://doi.org/10.5802/aif.1922
Classification:  32V25,  32V40,  32V10,  32D10,  32D20
Mots clés: principe de réflexion, principe de continuité, difféomorphisme CR, non dégénérescence holomorphe, minimalité globale au sens de Trépreau-Tumanov, fonction de réflexion, enveloppes d'holomorphie

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