Extension of holomorphic maps between real hypersurfaces of different dimension

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 .