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

Merker, Joel

HAL, hal-00003382 / Harvested from HAL

Résumé

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.

Détail

Publié le : 2002-07-05
Classification:  envelopes of holomorphy,  holomorphic nondegeneracy,  global minimality in the sense of Trépreau-Tumanov,  reflection function,  Reflection principle,  continuity principle,  CR diffeomorphism,  32V25, 32V40, 32V15, 32V10, 32D10, 32D20,  [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]