Levi-flat extensions from a part of the boundary

Nikolay Shcherbina; Giuseppe Tomassini

doi:10.1016/j.crma.2003.10.015

Complex Analysis

Levi-flat extensions from a part of the boundary

Presented by: Paul Malliavin

Nikolay Shcherbina ¹; Giuseppe Tomassini ²

¹ Department of Mathematics, University of Göteborg, 412 96 Göteborg, Sweden
² Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 699-703.

Abstract
Résumé

Let G be a bounded domain in $ℂ \times ℝ$ such that $G \times ℝ \subset ℂ^{2}$ is strictly pseudoconvex and U an open subset of bG. We define an open subset $Ω^{U}$ of $\bar{G}$ with the property $Ω^{U} \cap bG = U$ such that the following extension theorem holds true: for every ϕ∈C(U) there exist two functions $Φ^{\pm} \in C (Ω^{U})$ such that Φ^±|_U=ϕ and the graphs Γ(Φ^±) of Φ^± are Levi-flat over $Ω^{U} \cap G$ . Moreover, for each $Φ \in C (Ω^{U})$ such that Φ|_U=ϕ and Γ(Φ) is Levi-flat over $Ω^{U} \cap G$ one has Φ⁻⩽Φ⩽Φ⁺ on $Ω^{U}$ . We also show that if G is diffeomorphic to a 3-ball and U is the union of simply-connected domains each of which is contained either in the “upper” or in the “lower” part of bG (with respect to the u-direction), then $Ω^{U}$ is the maximal domain of Levi-flat extensions for some function ϕ∈C(U).

Soient G un domaine borné dans $ℂ \times ℝ$ tel que $G \times ℝ \subset ℂ^{2}$ soit strictement pseudoconvexe et U un sous-ensemble ouvert de bG. On définit un sous-ensemble ouvert $Ω^{U}$ de $\bar{G}$ avec la propriété $Ω^{U} \cap bG = U$ tel que le résultat suivant soit valable : pour toute fonction ϕ∈C(U) il existe deux fonctions $Φ^{\pm} \in C (Ω^{U})$ telles que Φ^±|_U=ϕ et les graphes Γ(Φ^±) de Φ^± soient Levi-plats sur $Ω^{U} \cap G$ . De plus, pour toute $Φ \in C (Ω^{U})$ telle que Φ|_U=ϕ et Γ(Φ) soit Levi-plat sur $Ω^{U} \cap G$ , on a Φ⁻⩽Φ⩽Φ⁺ sur $Ω^{U}$ . On démontre aussi que si G est difféomorphe à la boule et U est une réunion de domaines simplement connexes, chacun d'entre eux étant contenu soit dans la partie supérieure, soit dans la partie inférieure de bG (par rapport à la direction u), alors $Ω^{U}$ est le domaine maximal pour l'extension Levi-plate d'une certaine fonction ϕ∈C(U).

Received: 2003-04-01
Accepted: 2003-10-07
Published online: 2003-11-14

DOI: 10.1016/j.crma.2003.10.015

Author's affiliations:

Nikolay Shcherbina ¹; Giuseppe Tomassini ²

¹ Department of Mathematics, University of Göteborg, 412 96 Göteborg, Sweden
² Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

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     title = {Levi-flat extensions from a part of the boundary},
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Nikolay Shcherbina; Giuseppe Tomassini. Levi-flat extensions from a part of the boundary. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 699-703. doi : 10.1016/j.crma.2003.10.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.015/

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