2004/03/31 by Joel Merker
Mathematics · #math.CV #math.DG #msc:32V25 #msc:32V40 #msc:32V15 #msc:32V10 #msc:32D10 #msc:32D20
published as Ann. Inst. Fourier (Grenoble) 52 (2002), no. 5, 1443--1523 · 49 pages, 7 figures
arxiv created 2004/03/31 · arxiv updated 2009/12/01
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 Cinfty-smooth CR diffeomorphism h: M to M' between two globally minimal real analytic hypersurfaces in Cn (n >1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish that the reflection function Rh' associated to such a Cinfty-smooth CR diffeomorphism between two globally minimal hypersurfaces in Cn (n > 1) always extends holomorphically to a neighborhood of the graph of h in M× M', without any nondegeneracy condition on M'. This gives a new version of the Schwarz symmetry principle in several complex variables. Finally, as an appendix, we show that every Cinfty-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. This answers a conjecture, explicitely stated for Cinfty-smooth maps at p.~328 of the survey article: S.M. Baouendi, P. Ebenfelt and L.-P. Rothschild, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309--336. Importantly, notice that a much stronger result, valid for continuous maps, has been obtained by K. Diederich and S. Pinchuk in: Michigan Math. J. \bf 51 (2003), no. 1, 111--140; no.~3, 667--668.