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Comptes Rendus Mathématique
Volume 341, n° 6
pages 343-348 (septembre 2005)
Doi : 10.1016/j.crma.2005.07.012
Received : 16 June 2005 ;  accepted : 19 July 2005
On boundaries of Levi-flat hypersurfaces in  
Bords dʼhypersurfaces Levi-plates dans  
 

Pierre Dolbeault a , Giuseppe Tomassini b , Dmitri Zaitsev c
a Institut de mathématiques de Jussieu, université Paris 6, 175, rue du Chevaleret, 75013 Paris, France 
b Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy 
c School of Mathematics, Trinity College, Dublin 2, Ireland 

Abstract

Let S be a smooth 2-codimensional real compact submanifold of  ,  . We address the problem of finding a compact hypersurface M , with boundary S , such that   is Levi-flat. We prove the following theorem. Assume that (i) S is nonminimal at every CR point, (ii) every complex point of S is flat and elliptic and there exists at least one such point, (iii) S does not contain complex submanifolds of dimension  . Then there exists a Levi-flat  -subvariety   with negligible singularities and boundary   (in the sense of currents) such that the natural projection   restricts to a CR diffeomorphism between S and  . To cite this article: P. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

Soit S une sous-variété réelle, lisse. compacte, de codimension 2 de  ,  . On considère le problème de lʼexistence dʼune hypersurface compacte M , de bord S , telle que   soit Levi-plate. On démontre le théorème suivant : supposons que (i) S est non minimale en tout point CR, (ii) tout point complexe de S est plat et elliptique et il en existe un au moins, (iii) S ne contient aucune sous-variété complexe de dimension  . Alors il existe une sous-variété  , à singularités négligeables, avec bord   (au sens des courants) et telle quel la projection naturelle   donne un difféomorphisme CR entre S et  . Pour citer cet article : P. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.


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