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

Merker, Joël

doi:10.5802/aif.1922

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 ¹

¹ Université de Provence, CMI, 39 rue Joliot-Curie, 13453 Marweille cedex 13 (France)

Annales de l'Institut Fourier, Tome 52 (2002) no. 5, pp. 1443-1523.

Résumé
Abstract

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 $h : M \to M^{'}$ de classe $𝒞^{\infty}$ entre deux hypersurfaces analytiques réelles globalement minimales de $ℂ^{n}$ ( $n \geq 2$ ) est analytique réel en tout point de $M$ si $M^{'}$ est holomorphiquement non-dégénérée. Plus généralement, nous établissons que la fonction de réflexion $ℛ_{h}^{'}$ associée à un tel difféomorphisme CR de classe $𝒞^{\infty}$ entre deux hypersurfaces analytiques réelles globalement minimales se prolonge toujours holomorphiquement à un voisinage du graphe de $\bar{h}$ dans $M \times {\bar{M}}^{'}$ , sans aucune condition de non-dégénérescence sur $M^{'}$ . Cet énoncé fournit une nouvelle version du principe de réflexion de Schwarz en plusieurs variables complexes. Enfin, nous démontrons que toute application $h : M \to M^{'}$ de classe $𝒞^{\infty}$ 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 $h$ .

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 $𝒞^{\infty}$ -smooth CR diffeomorphism $h : M \to M^{'}$ between two globally minimal real analytic hypersurfaces in $ℂ^{n}$ ( $n \geq 2$ ) is real analytic at every point of $M$ if $M^{'}$ is holomorphically nondegenerate. More generally, we establish that the reflection function $ℛ_{h}^{'}$ associated to such a $𝒞^{\infty}$ -smooth CR diffeomorphism between two globally minimal hypersurfaces in $ℂ^{n}$ ( $n \geq 1$ ) always extends holomorphically to a neighborhood of the graph of $\bar{h}$ in $M \times {\bar{M}}^{'}$ , without any nondegeneracy condition on $M^{'}$ . This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every $𝒞^{\infty}$ - smooth CR mapping $h : M \to M^{'}$ between two real analytic hypersurfaces containing no complex curves is real analytic at every point of $M$ , without any rank condition on $h$ .

MR Zbl

DOI : 10.5802/aif.1922

Classification : 32V25, 32V40, 32V10, 32D10, 32D20
Keywords: reflection principle, continuity principle, CR diffeomorphism, holomorphic nondegeneracy, global minimality in the sense of Trépreau-Tumanov, reflection function, envelopes of holomorphy
Mot 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

Affiliations des auteurs :

Merker, Joël ¹

¹ Université de Provence, CMI, 39 rue Joliot-Curie, 13453 Marweille cedex 13 (France)

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Merker, Joël. On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle. Annales de l'Institut Fourier, Tome 52 (2002) no. 5, pp. 1443-1523. doi : 10.5802/aif.1922. https://aif.centre-mersenne.org/articles/10.5802/aif.1922/

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