Geometric properties of mappings between hypersurfaces in complex space
type: Item
instance of: scholarly article
title: Geometric properties of mappings between hypersurfaces in complex space
zbMATH Open document ID: 0702.32014
author: Linda Preiss Rothschild
author: Mohamed Salah Baouendi
published in: Journal of Differential Geometry
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32C05
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 4150644
zbMATH Keywords: essential type
zbMATH Keywords: real analytic hypersurfaces in \({\mathbb{C}}^{n+1}\)
zbMATH Keywords: finite type
zbMATH Keywords: multiplicity
zbMATH Keywords: \({\mathbb{C}}R\) mappings
zbMATH Keywords: holomorphic extendability
reviewed by: Stancho Dimiev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214444323
OpenAlex ID: W1489575973
Wikidata QID: Q115182252
DOI: 10.4310/JDG/1214444323
description: scientific article; zbMATH DE number 4150644