Condition R and holomorphic mappings of domains with generic corners

Abstract

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate.