Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions

Maryam Al-Towailb; Nabil Ourimi

doi:10.1016/j.crma.2012.08.008

Complex Analysis

Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions

the Editorial Board

Maryam Al-Towailb ¹; Nabil Ourimi ¹

¹Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 671-675

Abstract (Original language)
Résumé

Let D, $D^{'}$ be arbitrary domains in $C^{n}$ and $C^{N}$ respectively, $1 < n ⩽ N$ , both possibly unbounded and let $M \subset \partial D$ , $M^{'} \subset \partial D^{'}$ be open pieces of the boundaries. Suppose that ∂D is smooth real-analytic and minimal in an open neighborhood of $\bar{M}$ and $\partial D^{'}$ is smooth real-algebraic and minimal in an open neighborhood of ${\bar{M}}^{'}$ . Let $f : D \to D^{'}$ be a holomorphic mapping. Assume that the cluster set ${cl}_{f} (M)$ does not intersect $D^{'}$ . It is proved that if the cluster set ${cl}_{f} (p)$ of a point $p \in M$ contains some point $q \in M^{'}$ and the graph of f extends as an analytic set to a neighborhood of $(p, q) \in C^{n} \times C^{N}$ , then f extends as a holomorphic map near p.

Soient D, $D^{'}$ deux domaines respectivement de $C^{n}$ et $C^{N}$ , $1 < n ⩽ N$ et soient $M \subset \partial D$ , $M^{'} \subset \partial D^{'}$ deux parties ouvertes des frontières. Supposons que ∂D (resp. $\partial D^{'}$ ) est lisse, minimale et analytique réelle dans un voisinage de $\bar{M}$ (resp. lisse, minimale et algébrique réelle dans un voisinage de ${\bar{M}}^{'}$ ). Soit $f : D \to D^{'}$ une application holomorphe telle que lʼensemble des points limites ${cl}_{f} (M)$ nʼintersecte pas $D^{'}$ . Nous montrons que si lʼensemble des points limites ${cl}_{f} (p)$ dʼun point $p \in M$ contient un point $q \in M^{'}$ et le graphe de f se prolonge comme un ensemble analytique dans un voisinage de $(p, q) \in C^{n} \times C^{N}$ , alors f se prolonge holomorphiquement dans un voisinage de p.

Article information
Cite this article

Received: 2012-05-15
Accepted: 2012-08-29
Published online: 2012-07-01

DOI: 10.1016/j.crma.2012.08.008

Author's affiliations:

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

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

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

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