On boundaries of Levi-flat hypersurfaces in $ {\mathbb{C}}^{n}$

Pierre Dolbeault; Giuseppe Tomassini; Dmitri Zaitsev

doi:10.1016/j.crma.2005.07.012

Complex Analysis

On boundaries of Levi-flat hypersurfaces in

C^{n}

Presented by: Paul Malliavin

Pierre Dolbeault ¹; Giuseppe Tomassini ²; Dmitri Zaitsev 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

Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 343-348.

Abstract
Résumé

Let S be a smooth 2-codimensional real compact submanifold of $C^{n}$ , $n > 2$ . We address the problem of finding a compact hypersurface M, with boundary S, such that $M ∖ S$ 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 $n - 2$ . Then there exists a Levi-flat $(2 n - 1)$ -subvariety $\tilde{M} \subset C \times C^{n}$ with negligible singularities and boundary $\tilde{S}$ (in the sense of currents) such that the natural projection $π : C \times C^{n} \to C^{n}$ restricts to a CR diffeomorphism between S and $\tilde{S}$ .

Soit S une sous-variété réelle, lisse. compacte, de codimension 2 de $C^{n}$ , $n > 2$ . On considère le problème de l'existence d'une hypersurface compacte M, de bord S, telle que $M ∖ S$ 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 $n - 2$ . Alors il existe une sous-variété $\tilde{M} \subset C \times C^{n}$ , à singularités négligeables, avec bord $\tilde{S}$ (au sens des courants) et telle quel la projection naturelle $π : C \times C^{n} \to C^{n}$ donne un difféomorphisme CR entre S et $\tilde{S}$ .

Received: 2005-06-16
Accepted: 2005-07-19
Published online: 2005-09-02

DOI: 10.1016/j.crma.2005.07.012

Author's affiliations:

Pierre Dolbeault ¹; Giuseppe Tomassini ²; Dmitri Zaitsev 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

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     author = {Pierre Dolbeault and Giuseppe Tomassini and Dmitri Zaitsev},
     title = {On boundaries of {Levi-flat} hypersurfaces in $ {\mathbb{C}}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {343--348},
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     volume = {341},
     number = {6},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.012},
     language = {en},
}

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Pierre Dolbeault; Giuseppe Tomassini; Dmitri Zaitsev. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.012/

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