Analytic continuation of holomorphic mappings

Ayed, Besma; Ourimi, Nabil

doi:10.1016/j.crma.2009.07.001

Complex Analysis

Analytic continuation of holomorphic mappings
[Continuation analytique d'applications holomorphes]

Présenté par : 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, Tome 347 (2009) no. 17-18, pp. 1011-1016.

Résumé
Abstract

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.

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.

Reçu le : 2009-04-13
Accepté le : 2009-06-24
Publié le : 2009-07-17

DOI : 10.1016/j.crma.2009.07.001

Affiliations des auteurs :

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

@article{CRMATH_2009__347_17-18_1011_0,
     author = {Ayed, Besma and Ourimi, Nabil},
     title = {Analytic continuation of holomorphic mappings},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1011--1016},
     publisher = {Elsevier},
     volume = {347},
     number = {17-18},
     year = {2009},
     doi = {10.1016/j.crma.2009.07.001},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2009.07.001/}
}

TY  - JOUR
AU  - Ayed, Besma
AU  - Ourimi, Nabil
TI  - Analytic continuation of holomorphic mappings
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 1011
EP  - 1016
VL  - 347
IS  - 17-18
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2009.07.001/
DO  - 10.1016/j.crma.2009.07.001
LA  - en
ID  - CRMATH_2009__347_17-18_1011_0
ER  -

%0 Journal Article
%A Ayed, Besma
%A Ourimi, Nabil
%T Analytic continuation of holomorphic mappings
%J Comptes Rendus. Mathématique
%D 2009
%P 1011-1016
%V 347
%N 17-18
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2009.07.001/
%R 10.1016/j.crma.2009.07.001
%G en
%F CRMATH_2009__347_17-18_1011_0

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

Bibliographie
Cité par

[1] Baouendi, M.S.; Rotshschild, L.P. Images of real hypersurfaces under holomorphic mappings, J. Differential Geom., Volume 36 (1992), pp. 75-88

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

[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. Regularity of continuous CR maps in arbitrary dimension, Michigan Math. J., Volume 51 (2003) no. 1, pp. 111-140

[5] Huang, X. Schwarz reflection principle in complex spaces of dimension 2, Comm. Partial Differential Equations, Volume 21 (1996) no. 11–12, pp. 1781-1828

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

[7] Pinchuk, S.; Verma, K. Analytic sets and the boundary regularity of CR-mappings, Proc. AMS, Volume 129 (2001), pp. 2623-2632

[8] Shafikov, R. Analytic continuation of germs of holomorphic mappings, Michigan Math. J., Volume 47 (2000), pp. 133-149

[9] Shafikov, R. On boundary regularity of proper holomorphic mappings, Math. Z., Volume 242 (2002), pp. 517-528

[10] Shafikov, R.; Verma, K. A local extension theorem for proper holomorphic mappings in $C^{2}$ , J. Geom. Anal., Volume 13 (2003) no. 4, pp. 697-714

[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

Cité par Sources :