Holomorphic correspondences between CR manifolds

Analytic Continuation of Holomorphic Mappings from Nonminimal Hypersurfaces

“…It follows then that a Levi nondegenerate hypersurface M is (k, l)-spherical at one point iff it has this property at all points and we simply call M a (k, l)-spherical hypersurface. Similar extension result holds if instead of Levi nondegeneracy one assumes that M is essentially finite, a condition on the so-called Segre map of M generalizing Levi nondegeneracy, see [23] and [12]. Using arguments similar to those in [12] one can further generalize Pinchuk's theorem to the case when M is merely minimal in the sense of Tumanov [26], i.e., when M does not contain any germs of complex hypersurfaces, see [21].…”

“…Let M be a connected smooth real analytic hypersurface in C n , n > 1. It was shown by S. Pinchuk for Q = S 2n−1 [18], and by D. Hill and the second author [12] for the general case that if M is Levi nondegenerate then a germ of a local biholomorphic map f : M → Q extends locally biholomorphically along any path on M with the extension sending M to Q. This leads to the following definition: a Levi nondegenerate hypersurface M is called (k, l)-spherical at a point p ∈ M if there exists a germ at p of a biholomorphic map f sending the germ (M, p) of M at p onto the germ of a (k, l)-hyperquadric Q at f (p).…”

“…Example 7.3. For the hypersurfaces M α ⊂ C 2 given by (12) with α ∈ Z and the hypersurfaces K m ⊂ C 2 given by (13), the monodromy operator is the identity, and the map F is single-valued. For the hypersurfaces M α with α / ∈ Z the monodromy operator has a diagonal Jordan normal form:…”

Uniformization of strictly pseudoconvex domains. I

“…Once again, the case of a simply connected M has been treated previously in [13]. Incidentally, the methods of that paper can be used to generalize Theorem B to generic pseudoconcave CR-submanifolds of higher codimension in CP n with locally injective Segre maps.…”

“…Hence, we obtain the following (cp. [14] and [13]): Corollary 2.10. Let (D, p) be a domain over CP n such that every holomorphic function on D is constant.…”