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
Asian Journal of Mathematics 7 4
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.
Please note: There is a known bug in some browsers that causes an
error when a user tries to view large pdf file within the browser window.
If you receive the message "The file is damaged and could not be
repaired", please try one of the solutions linked below based on the
browser you are using.
Items in TARA are protected by copyright, with all rights reserved, unless otherwise indicated.