Analytic regularity of CR-mappings

Images of real submanifolds under finite holomorphic mappings

“…This study is partly motivated by the recent interest in the structure of nondegenerate mappings (e.g., finite holomorphic mappings) taking one real-analytic submanifold in C N into another. We mention here only the papers [4,5,9,10,12,13], and refer the reader to these papers for precise results and a more extensive bibliography.…”

Analyticity of Smooth CR Mappings of Generic Submanifolds

“…There are also many subsequent results implying analyticity of a smooth CR mapping when the targetX is a real-analytic generic submanifold M of C N ′ under various hypotheses on M and f . We mention here only, in addition to [MMZ02], the works [DW80], [Han83], [BJT85], [BR88], [DF88], [F89], [Pu90], [BHR96], [H96], [CPS00], [D01], [MMZ03b] and refer the reader to the bibliographies of these for further references. Previous results on convergence of formal mappings were given e.g.…”

Analytic regularity of CR maps into spheres

“…Moreover since M is minimal at 0, f extends to a holomorphic map F defined on one side of M near the origin [BT84,Tr86]. Therefore by using this extension F , it is easy to see that any ratio in the class R N −N +1 can be written in the form (2.1.1) for a suitable Ψ, Φ and G (for further details on that matter see [MMZ02,Lemma 6.1]). Since each component of f is CR, we conclude from Theorem 2.6 that each such component extends meromorphically to a neighborhood of the origin in C N .…”

“…The main novelty of this paper consists in the proof of the above mentioned meromorphic extension of f regardless of the codimension N − N (see Proposition 2.1). The ingredients of the proof rely on another meromorphic extension result for a class of CR ratios proved in [MMZ02], and an inductive dichotomy (in Lemma 2.8 below) showing that necessarily each component of the map f belongs to this class of ratios (see also [M02] for a related argument in the one-codimensional case in another context).…”