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

Joel Merker (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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 -> M' between two globally minimal real analytic hypersurfaces in C^n (n > 1) 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 always extends holomorphically to a neighborhood of the graph of \bar h in M \times \overline 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.

Mots clés: Reflection principle; continuity principle; CR diffeomorphism; holomorphic nondegeneracy; global minimality in the sense of Trépreau-Tumanov; reflection function; envelopes of holomorphy

Domaines: Géométrie différentielle [math.DG]

Joel Merker. On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle. Annales de l'Institut Fourier, 2002, 52, pp.1443-1523. ⟨hal-00003382⟩