Extension of holomorphic maps between real hypersurfaces of different dimension

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 Cⁿ, 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 CR-curve on M with the extension sending M to M′. Further, if D and D′ are smoothly bounded domains in Cⁿ 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.