Complex Analysis
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.

Let D, $D′$ be arbitrary domains in $Cn$ and $CN$ respectively, $1, both possibly unbounded and let $M⊂∂D$, $M′⊂∂D′$ 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 $f:D→D′$ 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 $p∈M$ contains some point $q∈M′$ 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.

Soient D, $D′$ deux domaines respectivement de $Cn$ et $CN$, $1 et soient $M⊂∂D$, $M′⊂∂D′$ 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 $f:D→D′$ 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 $p∈M$ contient un point $q∈M′$ 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.

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