## Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

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**Citation: **

*Meylan, Francine; Mir, Nordine; Zaitsev, Dmitri 'Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds' in Asian Journal of Mathematics, 7, (4), 2003, pp 493 - 509*

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**Abstract: **

The classical Schwarz reflection principle states that a continuous map f between real-analytic
curves M and M? in C that locally extends holomorphically to one side of M, extends also
holomorphically to a neighborhood of M in C. It is well-known that the higher-dimensional
analog of this statement for maps f : M ? M? between real-analytic CR-submanifolds M ? CN
andM? ? CN? does not hold without additional assumptions (unless M andM? are totally real). In
this paper, we assume that f is C?-smooth and that the target M? is real-algebraic, i.e. contained
in a real-algebraic subset of the same dimension. If f is known to be locally holomorphically
extendible to one side of M (when M is a hypersurface) or to a wedge with edge M (when M is a
generic submanifold of higher codimension), then f automatically satisfies the tangential Cauchy-
Riemann equations, i.e. it is CR. On the other hand, if M is minimal, any CR-map f : M ? M?
locally extends holomorphically to a wedge with edge M by Tumanov?s theorem [Tu88] and
hence, in that case, the extension assumption can be replaced by assuming f to be CR.

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**Author's Homepage: **

http://people.tcd.ie/zaitsevd
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**Description: **

PUBLISHED
**Author:**ZAITSEV, DMITRI

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**Publisher: **

International Press
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**Type of material: **

Journal Article#####
**Series/Report no: **

Asian Journal of Mathematics7

4

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**Availability: **

Full text available#####
**Keywords: **

Pure & Applied Mathematics
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