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

Merker, Joël

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

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 $h:M\to M^{\text{'}}$ de classe ${𝒞}^{\infty }$ entre deux hypersurfaces analytiques réelles globalement minimales de ${ℂ}^n$ ($n\ge 2$) est analytique réel en tout point de $M$ si $M^{\text{'}}$ est holomorphiquement non-dégénérée. Plus généralement, nous établissons que la fonction de réflexion ${ℛ}_h^{\text{'}}$ 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 $\overline{h}$ dans $M\times {\overline{M}}^{\text{'}}$, sans aucune condition de non-dégénérescence sur $M^{\text{'}}$. Cet énoncé fournit une nouvelle version du principe de réflexion de Schwarz en plusieurs variables complexes. Enfin, nous démontrons que toute application $h:M\to M^{\text{'}}$ 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 $h:M\to M^{\text{'}}$ between two globally minimal real analytic hypersurfaces in ${ℂ}^n$ ($n\ge 2$) is real analytic at every point of $M$ if $M^{\text{'}}$ is holomorphically nondegenerate. More generally, we establish that the reflection function ${ℛ}_h^{\text{'}}$ associated to such a ${𝒞}^{\infty }$-smooth CR diffeomorphism between two globally minimal hypersurfaces in ${ℂ}^n$ ($n\ge 1$) always extends holomorphically to a neighborhood of the graph of $\overline{h}$ in $M\times {\overline{M}}^{\text{'}}$, without any nondegeneracy condition on $M^{\text{'}}$. This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every ${𝒞}^{\infty }$- smooth CR mapping $h:M\to M^{\text{'}}$ between two real analytic hypersurfaces containing no complex curves is real analytic at every point of $M$, without any rank condition on $h$.

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