Condition R and holomorphic mappings of domains with generic corners

TL;DR

This paper investigates the extension properties of biholomorphic and proper maps between domains with generic corners, establishing conditions under which these maps extend smoothly to the boundary and characterizing boundary degeneracy.

Contribution

It proves that biholomorphic maps between domains with generic corners extend smoothly if and only if the target domain also has generic corners and satisfies Condition R, and characterizes proper maps to product domains.

Findings

Biholomorphic maps extend smoothly under specified conditions.

Proper maps extend continuously and smoothly on boundary parts.

Boundary degeneracy is necessary for certain proper maps.

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.