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A generalized complex Hopf lemma and its applications to CR mappings. (English) Zbl 0781.32021

A new approach to the theory of smooth \(\mathbb{C}\mathbb{R}\)-mappings between hypersurfaces in a complex vector space is developed in this very interesting paper. It is based on the techniques of analytic discs and the new notion of minimal convexity at a point introduced by the authors. At the bottom is the following observation: when the target hypersurface is minimally convex at a point then the derivative in the transversal direction of the transversal component of the mapping does not vanish at this point. (This assertion is to be compared with the classical Hopf lemma about harmonic functions!) The notion of minimal convexity at a point is closely related with the Tumanov’s notion of minimality at a point, namely this is the requirement of the local existence of a side of the tangent space in this point such that the real derivative of each analytic disc of sufficient small norm passing through this point lies in that side or in the mentioned tangent space. In the case of a smooth pseudoconvex hypersurface the minimality in a point implies minimal convexity in the same point. The main authors’ result is that any smooth \(\mathbb{C}\mathbb{R}\) self-mapping of a hypersurface of \(D\)-finite type (D’Angelo) fixing a point is either constant or local diffeomorphism. For real analytic hyersurfaces the obtained results, combined with some previous ones of the authors, give new theorems for holomorphic extendibility and local biholomorphisms of \(CR\)-mappings.
Reviewer: S.Dimiev (Sofia)

MSC:

32V05 CR structures, CR operators, and generalizations
32T99 Pseudoconvex domains
32D15 Continuation of analytic objects in several complex variables
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References:

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