It is proved that the germ of a holomorphic map from a real analytic
hypersurface M in C^n into a strictly pseudoconvex compact real algebraic
hypersurface M' in C^N, 1 < n < N extends holomorphically along any path on M

Shafikov, Rasul

Verma, Kaushal

Annales de l’institut Fourier

English

arXiv

Verma, Kausha

Annales de l’institut Fourier (AIF)

Extension of holomorphic maps between real hypersurfaces of different dimension

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 Cn, M′ be a compact strictly pseudoconvex real algebraic hypersurface in CN, 1 &#60; n &#8804; 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 Cn and CN respectively, 1 &#60; n &#8804; N, the boundary of D is real analytic, and the boundary of D′ is real algebraic, and if f : D &#8594; 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

Indian Academy of Sciences

Publications of the IAS Fellows

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 $D$

Name not available

Numérisation de Documents Anciens Mathématiques

Rasul Shafikov

Kausha Verma

Crossref

Open Access Repository of IISc Research Publications

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.9437

Extension of holomorphic maps between real hypersurfaces of different
  dimension

Extension of holomorphic maps between real hypersurfaces of different dimension

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