Proper holomorphic mappings between real-analytic pseudoconvex domains inC n

Diederich, Klas; Fornæss, John Erik

doi:10.1007/bf01462892

Cited by 93 publications

(91 citation statements)

References 20 publications

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“…As it was mentioned in the introduction, Webster was the the first to use Segre varieties in the context of holomorphic mappings. His ideas were further developed in [DW], [DF2], [DFY], [DP1] and other papers. Our main construction is based on the technique of these papers.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Holomorphic correspondences between CR manifolds

Hill¹,

Shafikov²

2005

Indiana Univ. Math. J.

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Section: Statement Of Resultsmentioning

confidence: 99%

Holomorphic correspondences between CR manifolds

Hill¹,

Shafikov²

2005

Indiana Univ. Math. J.

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“…There are also many subsequent results implying analyticity of a smooth CR mapping when the targetX is a real-analytic generic submanifold M of C N ′ under various hypotheses on M and f . We mention here only, in addition to [MMZ02], the works [DW80], [Han83], [BJT85], [BR88], [DF88], [F89], [Pu90], [BHR96], [H96], [CPS00], [D01], [MMZ03b] and refer the reader to the bibliographies of these for further references. Previous results on convergence of formal mappings were given e.g.…”

Section: The Formal Mapping H Is Said To Be Finite If the Ideal I(h) mentioning

confidence: 99%

Analyticity of Smooth CR Mappings of Generic Submanifolds

Ebenfelt¹,

Rothschild²

2007

Asian Journal of Mathematics

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Abstract. We consider a smooth CR mapping f from a real-analytic generic submanifold M in C N into C N . For M of finite type and essentially finite at a point p ∈ M , and f formally finite at p, we give a necessary and sufficient condition for f to extend as a holomorphic mapping in some neighborhood of p. In a similar vein, we consider a formal holomorphic mapping H and give a necessary and sufficient condition for H to be convergent.

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“…Following [DP1] we call U 1 and U 2 a standard pair of neighborhoods of z 0 . A detailed discussion of elementary properties of Segre varieties can be found in [DW], [DF2], [DP1] or [BER2]. The map λ : z → Q z is called the Segre map.…”

Section: Background Materialmentioning

confidence: 99%

Analytic continuation of holomorphic correspondences and equivalence of domains in ℂ n

Shafikov¹

2003

Invent. math.

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The following result is proved: Let D and D ′ be bounded domains in C n , ∂D is smooth, real-analytic, simply connected, and ∂D ′ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence f : D → D ′ if and only if ∂D and ∂D ′ a locally CR-equivalent. This leads to a characterization of the equivalence relationship between bounded domains in C n modulo proper holomorphic correspondences in terms of CR-equivalence of their boundaries.1 Definitions and Main Results. In this paper we address the following question: Given two domains D and D ′ , when does there exist a proper holomorphic correspondence F : D → D ′ ? Note (see [St2]) that the existence of a correspondence F defines an equivalence relation D ∼ D ′ . This equivalence relation is a natural generalization of biholomorphic equivalence of domains in C n .To illustrate the concept of equivalence modulo holomorphic correspondences, consider domains of the form Ω p,q = {|z 1 | 2p + |z 2 | 2q < 1}, p, q ∈ Z + . Then f (z) = (z 1 p/s , z 2 q/t ) is a proper holomorphic correspondence between Ω p,q and Ω s,t , while a proper holomorphic map from Ω p,q to Ω s,t exists only if s|p and t|q, or s|q and t|p. For details see [Bd] or [La].The main result of this paper is the following theorem.

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Proper holomorphic mappings between real-analytic pseudoconvex domains inC n

Cited by 93 publications

References 20 publications

Holomorphic correspondences between CR manifolds

Holomorphic correspondences between CR manifolds

Analyticity of Smooth CR Mappings of Generic Submanifolds

Analytic continuation of holomorphic correspondences and equivalence of domains in ℂ n

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