Comptes Rendus
Complex Analysis
Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions
[Ensembles analytiques prolongeant les graphes dʼapplications holomorphes entre domaines de dimensions différentes]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 671-675.

Soient D, D deux domaines respectivement de Cn et CN, 1<nN et soient MD, MD deux parties ouvertes des frontières. Supposons que ∂D (resp. D) est lisse, minimale et analytique réelle dans un voisinage de M¯ (resp. lisse, minimale et algébrique réelle dans un voisinage de M¯). Soit f:DD une application holomorphe telle que lʼensemble des points limites clf(M) nʼintersecte pas D. Nous montrons que si lʼensemble des points limites clf(p) dʼun point pM contient un point qM et le graphe de f se prolonge comme un ensemble analytique dans un voisinage de (p,q)Cn×CN, alors f se prolonge holomorphiquement dans un voisinage de p.

Let D, D be arbitrary domains in Cn and CN respectively, 1<nN, both possibly unbounded and let MD, MD be open pieces of the boundaries. Suppose that ∂D is smooth real-analytic and minimal in an open neighborhood of M¯ and D is smooth real-algebraic and minimal in an open neighborhood of M¯. Let f:DD be a holomorphic mapping. Assume that the cluster set clf(M) does not intersect D. It is proved that if the cluster set clf(p) of a point pM contains some point qM and the graph of f extends as an analytic set to a neighborhood of (p,q)Cn×CN, then f extends as a holomorphic map near p.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.08.008

Maryam Al-Towailb 1 ; Nabil Ourimi 1

1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
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Maryam Al-Towailb; Nabil Ourimi. Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 671-675. doi : 10.1016/j.crma.2012.08.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.008/

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Cité par Sources :

The project was supported by the Research Center, College of Science, King Saud University.

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