On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle
| dc.creator | Merker, Joel | |
| dc.date | 2004-03-31 | |
| dc.date.accessioned | 2026-07-07T05:06:55Z | |
| dc.date.available | 2026-07-07T05:06:55Z | |
| dc.description | 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' 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 (n > 1) always extends holomorphically to a neighborhood of the graph of \bar{h} in M\times \bar{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 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. This answers a conjecture, explicitely stated for C^infty-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. | |
| dc.description | 49 pages, 7 figures | |
| dc.identifier | https://arxiv.org/abs/math/0403539 | |
| dc.identifier | http://arxiv.org/abs/math/0403539 | |
| dc.identifier | Ann. Inst. Fourier (Grenoble) 52 (2002), no. 5, 1443--1523 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70660 | |
| dc.subject | Complex Variables | |
| dc.subject | Differential Geometry | |
| dc.subject | 32V25, 32V40, 32V15, 32V10, 32D10, 32D20 | |
| dc.title | On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle | |
| dc.type | text |