The purpose of this paper is to organize some results on the local geometry
of CR singular real-analytic manifolds that are images of CR manifolds via a CR
map that is a diffeomorphism onto its image. We find a necessary (sufficient in
dimension 2) condition for the diffeomorphism to extend to a finite holomorphic
map. The multiplicity of this map is a biholomorphic invariant that is
precisely the Moser invariant of the image when it is a Bishop surface with
vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR
singular images and we prove that the set of CR singular points must be large,
and in the case of codimension 2, necessarily Levi-flat or complex. We also
show that there exist real-analytic CR functions on such images that satisfy
the tangential CR conditions at the singular points, yet fail to extend to
holomorphic functions in a neighborhood. We provide many examples to illustrate
the phenomena that arise.Comment: 21 pages, accepted to Arkiv for Mathemati
It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding f of a strictly psedoconvex hypersurface M 2n+1 ⊂ C n+1 into the sphere S 2N +1 ⊂ C N +1 is rigid, i.e. any other such local embedding is obtained from f by composition by an automorphism of the target sphere S 2N +1 , provided that the codimension N − n < n/2. In this paper, we consider the limit case N − n = n/2 in the simplest situation where n = 2, i.e. we consider local CR embeddings f : M 5 → S 7 . We show that there are at most two different local embeddings, up to composition with an automorphism of S 7 . We also identify a subclass of 5-dimensional, strictly pseudoconvex hypersurfaces M 5 in terms of their CR curvatures such that rigidity holds for local CR embeddings f : M 5 → S 7 .
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