Extension of holomorphic maps between real hypersurfaces of different dimension

Shafikov, Rasul; Verma, Kausha

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

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 $M^{\prime }$ une hypersurface compacte, strictement pseudoconvexe, et algébrique réelle dans $C^N$, avec $1<n\le N$. Supposons que $f$ soit le germe d’une application holomorphe en un point $p$ de $M$, et $f\left(M\right)$ soit contenu dans $M^{\prime }$. Alors $f$ se prolonge à un application holomorphe le long de toute courbe $CR$ sur $M$, et le prolongement envoie $M$ dans $M^{\prime }$. De plus, si $D$ et $D^{\prime }$ sont des domaines bornés lisses dans $C^n$ et $C^N$ respectivement, avec $1<n\le N$, la frontière de $D$ est analytique réelle, celle de D’ est algébrique réelle, et si $f:D\to D^{\prime }$ 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 $C^n$, $M^{\prime }$ be a compact strictly pseudoconvex real algebraic hypersurface in $C^N$, $1<n\le N$. Suppose that $f$ is a germ of a holomorphic map at a point $p$ in $M$ and $f\left(M\right)$ is in $M^{\prime }$. Then f extends as a holomorphic map along any smooth $CR$-curve on M with the extension sending $M$ to $M^{\prime }$. Further, if $D$ and $D^{\prime }$ are smoothly bounded domains in $C^n$ and $C^N$ respectively, $1<n\le N$, the boundary of $D$ is real analytic, and the boundary of $D^{\prime }$ is real algebraic, and if $f:D\to D^{\prime }$ 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 $D$.

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

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