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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
##### MSC:
 58C10 Holomorphic maps on manifolds 32H99 Holomorphic mappings and correspondences 32G99 Deformations of analytic structures
##### Keywords:
holomorphic extensions; CR manifolds; smooth CR mapping
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