# Extension of holomorphic maps between real hypersurfaces of different dimension

Shafikov, Rasul and Verma, Kausha (2007) Extension of holomorphic maps between real hypersurfaces of different dimension. In: Annales de l'institut Fourier, 57 (6). pp. 2063-2080.

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## Abstract

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 CR-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 $f:D\leftrightarrow D'$ 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 $D$.

Item Type: Journal Article http://aif.cedram.org/aif-bin/item?id=AI... Copyright of this article belongs to Annales de L'Institut Fourier. Division of Physical & Mathematical Sciences > Mathematics 31 Jul 2008 27 Aug 2008 13:40 http://eprints.iisc.ernet.in/id/eprint/15379