# Extension of holomorphic maps between real hypersurfaces of different dimension

Rasul Shafikov^{[1]}; Kausha Verma^{[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 (2007)

- Volume: 57, Issue: 6, page 2063-2080
- ISSN: 0373-0956

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topShafikov, Rasul, and Verma, Kausha. "Extension of holomorphic maps between real hypersurfaces of different dimension." Annales de l’institut Fourier 57.6 (2007): 2063-2080. <http://eudml.org/doc/10288>.

@article{Shafikov2007,

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 $C^n$, $M^\{\prime\}$ 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 $f(M)$ 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 $f : D \rightarrow D^\{\prime\}$ 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 $D$.},

affiliation = {University of Western Ontario Department of Mathematics London N6A 5B7 (Canada); Indian Institute of Science Department of Mathematics Bangalore 560012 (India)},

author = {Shafikov, Rasul, Verma, Kausha},

journal = {Annales de l’institut Fourier},

keywords = {Holomorphic mappings; reflection Principle; boundary regularity; analytic continuation; holomorphic mappings; reflection principle},

language = {eng},

number = {6},

pages = {2063-2080},

publisher = {Association des Annales de l’institut Fourier},

title = {Extension of holomorphic maps between real hypersurfaces of different dimension},

url = {http://eudml.org/doc/10288},

volume = {57},

year = {2007},

}

TY - JOUR

AU - Shafikov, Rasul

AU - Verma, Kausha

TI - Extension of holomorphic maps between real hypersurfaces of different dimension

JO - Annales de l’institut Fourier

PY - 2007

PB - Association des Annales de l’institut Fourier

VL - 57

IS - 6

SP - 2063

EP - 2080

AB - 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 $f(M)$ 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 $f : D \rightarrow D^{\prime}$ 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 $D$.

LA - eng

KW - Holomorphic mappings; reflection Principle; boundary regularity; analytic continuation; holomorphic mappings; reflection principle

UR - http://eudml.org/doc/10288

ER -

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