Analytic continuation of germs of holomorphic mappings between real hypersurfaces in Cn.

Holomorphic correspondences between CR manifolds

“…Pinchuk [Pi2] proved that a germ of a biholomorphic map f from a strictly pseudoconvex, real-analytic, non-spherical hypersurface M to a compact, strictly pseudoconvex, real-analytic hypersurface M ′ ⊂ C n extends holomorphically along any path on M . A similar result was shown in [Sh1] for the case when M is essentially finite, smooth, real-analytic and M ′ ⊂ C n is compact, real-algebraic and strictly pseudoconvex. Levi non-degeneracy of the target hypersurface ensures that the extended map is single valued.…”

Extension of holomorphic maps between real hypersurfaces of different dimension

“…Theorem 1.1 is a direct generalization of these results (although our methods are quite different). Further, in the case when dim M = dim M , Theorem 1.1 generalizes the result in [29], where the hypersurface M was assumed to be essentially finite, a stronger condition than minimality. Other related results also include various extensions obtained when both M and M are algebraic (see e.g.…”

“…In the proof of Theorem 1.3 we modify the approach in [29] to our situation. The strategy can be outlined as follows.…”

On Boundary Regularity of Holomorphic Correspondences

“…The ideas of the proof were given in [19]. Assume that Q 0 ⊂ E. From Proposition 4.1 of [18] there exists a point t ∈ Γ\E such that Q 0 ∩ Q t ̸ = ∅. Let h :Ũ → C n be the germ of the correspondence f defined in a neighborhoodŨ of t. We shrinkŨ and choose V in such a way that for any w ∈ V , the set Q w ∩Ũ is connected.…”