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

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.