Analytic continuation of holomorphic mappings

Ayed, Besma; Ourimi, Nabil

doi:10.1016/j.crma.2009.07.001

Complex Analysis

Analytic continuation of holomorphic mappings

Presented by: Jean-Pierre Demailly

Ayed, Besma ¹; Ourimi, Nabil ²

¹ Faculté des sciences de Monastir, route de Kairouan, Monastir, 5019, Tunisia
² King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1011-1016.

Abstract
Résumé

Let D be a domain in $C^{n}$ , $n > 1$ , and $f : D \to C^{n}$ be a holomorphic map. Let $U \subset C^{n}$ be an open set such that $M : = \partial D \cap U$ is in U a relatively closed, connected, smooth real-analytic hypersurface of finite type (in the sense of D'Angelo). Suppose that the cluster set ${cl}_{f} (M)$ is contained in a closed, smooth real-algebraic hypersurface $M^{'} \subset U^{'}$ of finite type, where $U^{'}$ is an open set in $C^{n}$ . It is shown that if f extends continuously to some open piece of M, then it extends holomorphically to a neighborhood of each point of M. Note that here the compactness of $M^{'}$ is not required.

Soient D un domaine de $C^{n}$ , $n > 1$ , et $f : D \to C^{n}$ une application holomorphe. Soit $U \subset C^{n}$ un ouvert tel que $M : = \partial D \cap U$ est une hypersurface relativement fermée dans U, connexe, lisse, analytique réelle et de type fini (au sens de D'Angelo). Supposons que l'ensemble des points limites ${cl}_{f} (M)$ est contenu dans une hypersurface, fermée, lisse, algébrique réelle $M^{'} \subset U^{'}$ de type fini, où $U^{'}$ est un ouvert de $C^{n}$ . Nous montrons que si f se prolonge continûment sur une partie ouverte de M, alors elle se prolonge holomorphiquement au voisinage de chaque point de M. Notons qu'ici la compacité de $M^{'}$ n'est pas exigée.

Received: 2009-04-13
Accepted: 2009-06-24
Published online: 2009-07-17

DOI: 10.1016/j.crma.2009.07.001

Author's affiliations:

Ayed, Besma ¹; Ourimi, Nabil ²

¹ Faculté des sciences de Monastir, route de Kairouan, Monastir, 5019, Tunisia
² King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

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     title = {Analytic continuation of holomorphic mappings},
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     pages = {1011--1016},
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Ayed, Besma; Ourimi, Nabil. Analytic continuation of holomorphic mappings. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1011-1016. doi : 10.1016/j.crma.2009.07.001. http://www.numdam.org/articles/10.1016/j.crma.2009.07.001/

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