| 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. |
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