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

Al-Towailb, Maryam; Ourimi, Nabil

doi:10.1016/j.crma.2012.08.008

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]

Présenté par : the Editorial Board

Al-Towailb, Maryam ¹ ; Ourimi, Nabil ¹

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

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

Résumé
Abstract

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.

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.

Reçu le : 2012-05-15
Accepté le : 2012-08-29
Publié le : 2012-07-01

DOI : 10.1016/j.crma.2012.08.008

Affiliations des auteurs :

Al-Towailb, Maryam ¹ ; Ourimi, Nabil ¹

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

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     title = {Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions},
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     pages = {671--675},
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Al-Towailb, Maryam; Ourimi, Nabil. Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions. Comptes Rendus. Mathématique, Tome 350 (2012) no. 13-14, pp. 671-675. doi : 10.1016/j.crma.2012.08.008. http://www.numdam.org/articles/10.1016/j.crma.2012.08.008/

Bibliographie
Cité par

[1] Ayed, B.; Ourimi, N. Analytic continuation of holomorphic mappings, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 1011-1016

[2] Coupet, B.; Damour, S.; Merker, J.; Sukhov, A. Sur lʼanalyticité des applications CR lisses à valeurs dans un ensemble algébrique réel, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002) no. 11, pp. 953-956

[3] Diederich, K.; Pinchuk, S. Proper holomorphic maps in dimension 2 extend, Indiana Univ. Math. J., Volume 44 (1995), pp. 1089-1126

[4] Diederich, K.; Pinchuk, S. Analytic sets extending the graphs of holomorphic mappings, J. Geom. Anal., Volume 14 (2004) no. 2, pp. 231-239

[5] Lojasiewicz, S. Introduction to Complex Analytic Geometry, Birkhäuser, Basel, 1991

[6] Merker, J. On the local geometry of generic submanifolds of $C^{n}$ and the analytic reflection principle (part I), J. Math. Sci., Volume 125 (2005) no. 6, pp. 751-824

[7] Merker, J.; Porten, E. On wedge extendability of CR-meromorphic functions, Math. Z., Volume 241 (2002), pp. 485-512

[8] Meylan, F.; Mir, N.; Zaitsev, D. Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds, Asian J. Math., Volume 7 (2003) no. 4, pp. 503-519

[9] Narasimhan, R. Introduction to the Theory of Analytic Spaces, Lecture Notes in Math., vol. 25, Springer-Verlag, New York, 1966

[10] Shafikov, R. Analytic continuation of germs of holomorphic mappings between real hypersurfaces in $C^{n}$ , Michigan Math. J., Volume 47 (2001) no. 1, pp. 133-149

[11] Shafikov, R.; Verma, K. Extension of holomorphic maps between real hypersurfaces of different dimensions, Ann. Inst. Fourier, Grenoble, Volume 57 (2007) no. 6, pp. 2063-2080

[12] Tumanov, A. Foliations by complex curves and the geometry of real surfaces of finite type, Math. Z., Volume 240 (2002), pp. 385-388

Cité par Sources :

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