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CR mappings and their holomorphic extension. (English) Zbl 0639.58002
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1987, Exp. No. 23, 6 p. (1987).
The authors study CR manifolds, in particular hypersurfaces in \({\mathbb{C}}^{n+1}\). If M, M’ are real analytic hypersurfaces (of real dimension \(2n+1)\) in \({\mathbb{C}}^{n+1}\), if \(H: M\to M'\) is a smooth CR mapping, defined near \(p_ 0\in M\) with \(Hp_ 0=p_ 0'\), if H is “generic” at \(p_ 0\) \([i.e.\quad H'({\mathbb{C}}T_{p_ 0}M)\not\subseteq \nu '_{p_ 0'}\oplus {\bar \nu}'_{p_ 0'}],\) and if M, M’ are “essentially finite” at \(p_ 0\), \(p_ 0'\), then H extends to a holomorphic map from a neighborhood of \(p_ 0\) in \({\mathbb{C}}^{n+1}\) to \({\mathbb{C}}^{n+1}\). This is used to prove an extension theorem for domains in \({\mathbb{C}}^{n+1}:\) given a proper holomorphic map \(H: D\to D'\) for D, D’ bounded pseudoconvex domains in \({\mathbb{C}}^{n+1}\) with real analytic boundaries (n\(\geq 1)\), then H extends holomorphically to a neighborhood of the closure of D. Finally, the case of CR submanifolds of higher codimension is mentioned.
Reviewer: A.Aeppli
58C10 Holomorphic maps on manifolds
32H99 Holomorphic mappings and correspondences
32G99 Deformations of analytic structures
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