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

Abstract

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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 ≤ 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 ≤ N , the boundary of D is real analytic, and the boundary of D ′ is real algebraic, and if f : D → 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 .

How to cite

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Shafikov, 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 $C^N$, $1 &lt; n \le N$. 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, $1 &lt; n \le N$, 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 $C^N$, $1 &lt; n \le N$. 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, $1 &lt; n \le N$, 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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