Comptes Rendus
Complex Analysis
Levi-flat extensions from a part of the boundary
Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 699-703.

Let G be a bounded domain in × such that G× 2 is strictly pseudoconvex and U an open subset of bG. We define an open subset Ω U of G ¯ with the property Ω U bG =U such that the following extension theorem holds true: for every ϕC(U) there exist two functions Φ ± C(Ω U ) such that Φ±|U=ϕ and the graphs Γ(Φ±) of Φ± are Levi-flat over Ω U G. Moreover, for each ΦC(Ω U ) such that Φ|U=ϕ and Γ(Φ) is Levi-flat over Ω U G one has ΦΦΦ+ 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 ϕC(U).

Soient G un domaine borné dans × tel que G× 2 soit strictement pseudoconvexe et U un sous-ensemble ouvert de bG. On définit un sous-ensemble ouvert Ω U de G ¯ avec la propriété Ω U bG =U tel que le résultat suivant soit valable : pour toute fonction ϕC(U) il existe deux fonctions Φ ± C(Ω U ) telles que Φ±|U=ϕ et les graphes Γ(Φ±) de Φ± soient Levi-plats sur Ω U G. De plus, pour toute ΦC(Ω U ) telle que Φ|U=ϕ et Γ(Φ) soit Levi-plat sur Ω U G, on a ΦΦΦ+ 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 ϕC(U).

Published online:
DOI: 10.1016/j.crma.2003.10.015

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
     author = {Nikolay Shcherbina and Giuseppe Tomassini},
     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.

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