Complex Analysis
On boundaries of Levi-flat hypersurfaces in Cn
Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 343-348.

Let S be a smooth 2-codimensional real compact submanifold of Cn, n>2. We address the problem of finding a compact hypersurface M, with boundary S, such that MS 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 n2. Then there exists a Levi-flat (2n1)-subvariety M˜C×Cn with negligible singularities and boundary S˜ (in the sense of currents) such that the natural projection π:C×CnCn restricts to a CR diffeomorphism between S and S˜.

Soit S une sous-variété réelle, lisse. compacte, de codimension 2 de Cn, n>2. On considère le problème de l'existence d'une hypersurface compacte M, de bord S, telle que MS 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 n2. Alors il existe une sous-variété M˜C×Cn, à singularités négligeables, avec bord S˜ (au sens des courants) et telle quel la projection naturelle π:C×CnCn donne un difféomorphisme CR entre S et S˜.

Published online:
DOI: 10.1016/j.crma.2005.07.012
Dolbeault, Pierre 1; Tomassini, Giuseppe 2; Zaitsev, Dmitri 3

1 Institut de mathématiques de Jussieu, université Paris 6, 175, rue du Chevaleret, 75013 Paris, France
2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
3 School of Mathematics, Trinity College, Dublin 2, Ireland
     author = {Dolbeault, Pierre and Tomassini, Giuseppe and Zaitsev, Dmitri},
     title = {On boundaries of {Levi-flat} hypersurfaces in $ {\mathbb{C}}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {343--348},
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     year = {2005},
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Dolbeault, Pierre; Tomassini, Giuseppe; Zaitsev, Dmitri. On boundaries of Levi-flat hypersurfaces in $ {\mathbb{C}}^{n}$. Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 343-348. doi : 10.1016/j.crma.2005.07.012.

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