Extension of holomorphic maps between real hypersurfaces of different dimension

Shafikov, Rasul; Verma, Kausha

doi:10.5802/aif.2324

Shafikov, Rasul ¹ ; Verma, Kausha ²

¹ University of Western Ontario Department of Mathematics London N6A 5B7 (Canada)
² Indian Institute of Science Department of Mathematics Bangalore 560012 (India)

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2063-2080.

Résumé
Abstract

Dans cet article, les résultats sur le prolongement analytique des germes d’applications holomorphes d’une hypersurface analytique réelle à une hypersurface algébrique réelle sont étendus au cas où la cible est une hypersurface de dimension supérieure à celle de la source. Plus précisément, nous prouvons ce qui suit : soit $M$ une hypersurface lisse, connexe, analytique réelle et minimale dans $C^{n}$ , et $M^{'}$ une hypersurface compacte, strictement pseudoconvexe, et algébrique réelle dans $C^{N}$ , avec $1 < n \leq N$ . Supposons que $f$ soit le germe d’une application holomorphe en un point $p$ de $M$ , et $f (M)$ soit contenu dans $M^{'}$ . Alors $f$ se prolonge à un application holomorphe le long de toute courbe $C R$ sur $M$ , et le prolongement envoie $M$ dans $M^{'}$ . De plus, si $D$ et $D^{'}$ sont des domaines bornés lisses dans $C^{n}$ et $C^{N}$ respectivement, avec $1 < n \leq N$ , la frontière de $D$ est analytique réelle, celle de D’ est algébrique réelle, et si $f : D \to D^{'}$ est une application holomorphe propre qui admet un prolongement lisse à un voisinage d’un point $p$ de la frontière de $D$ , alors l’application $f$ se prolonge continûment à la fermeture de $D$ , et le prolongement est analytique sur un sous-ensemble dense de la frontière de $D$ .

In this paper we extend the results on analytic continuation of germs of holomorphic mappings from a real analytic hypersurface to a real algebraic hypersurface to the case when the target hypersurface is of higher dimension than the source. More precisely, we prove the following: Let $M$ be a connected smooth real analytic minimal hypersurface in $C^{n}$ , $M^{'}$ be a compact strictly pseudoconvex real algebraic hypersurface in $C^{N}$ , $1 < n \leq N$ . Suppose that $f$ is a germ of a holomorphic map at a point $p$ in $M$ and $f (M)$ is in $M^{'}$ . Then f extends as a holomorphic map along any smooth $C R$ -curve on M with the extension sending $M$ to $M^{'}$ . Further, if $D$ and $D^{'}$ are smoothly bounded domains in $C^{n}$ and $C^{N}$ respectively, $1 < n \leq N$ , the boundary of $D$ is real analytic, and the boundary of $D^{'}$ is real algebraic, and if $f : D \to D^{'}$ is a proper holomorphic map which admits a smooth extension to a neighbourhood of a point $p$ in the boundary of $D$ , then the map $f$ extends continuously to the closure of $D$ , and the extension is holomorphic on a dense open subset of the boundary of $D$ .

EuDML MR Zbl

DOI : 10.5802/aif.2324

Classification : 32H40
Keywords: Holomorphic mappings, reflection Principle, boundary regularity, analytic continuation
Mots clés : applications holomorphes, principe de réflexion, prolongement analytique

Affiliations des auteurs :

Shafikov, Rasul ¹ ; Verma, Kausha ²

¹ University of Western Ontario Department of Mathematics London N6A 5B7 (Canada)
² Indian Institute of Science Department of Mathematics Bangalore 560012 (India)

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     pages = {2063--2080},
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Shafikov, Rasul; Verma, Kausha. Extension of holomorphic maps between real hypersurfaces of different dimension. Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2063-2080. doi : 10.5802/aif.2324. https://aif.centre-mersenne.org/articles/10.5802/aif.2324/

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