Geometric properties of mappings between hypersurfaces in complex space (Q914901)

The mappings between hypersurfaces in complex vector space were studied by many authors in different categories (real analytic, holomorphic, \({\mathbb{C}}R)\). A general treatment for formal, smooth and real analytic hypersurfaces in \({\mathbb{C}}^{n+1}\) is developed in this paper. The study is based on the notions of finite type of a manifold (Kohn, Bloom- Graham), finite multiplicity of a mapping (authors) and essentially finiteness of a manifold (authors and Treves). The notions of a transversal coordinate for a manifold M and transversal component for a mapping H: \(M\to M'\) are introduced. The interplay between the above mentioned notions is completely described (see Th. 1, 2, 3). The obtained results are combined and applied to smooth \({\mathbb{C}}R\) mappings (see Th. 4, 5). The mentioned theorems generalize a result of Fornaess in the pseudoconvex case (Th. 1) and authors' results about \({\mathbb{C}}R\) mappings between real analytic manifolds (Th. 4, 5). The classical results of Pincuk and H. Lewy on holomorphic extendability in the strongly pseudoconvex case are generalized for smooth \({\mathbb{C}}R\) mappings between real analytic hypersurfaces. Proper holomorphic mappings between bounded domains D in \({\mathbb{C}}^{n+1}\) are studied too. The obtained result is used to relate global and local multiplicity of the mappings in D and on the boundary \(\partial D\) of D.