Condition R and proper holomorphic maps between equidimensional product domains

TL;DR

This paper proves that proper holomorphic maps between equidimensional product domains extend smoothly to the boundary, with the number of factors preserved, and the maps decompose into products of maps between factors, under Condition R.

Contribution

It establishes smooth boundary extension and factorization of proper holomorphic maps between product domains satisfying Condition R, revealing their structural properties.

Findings

Proper maps extend smoothly to the boundary under Condition R.

Number of factors in source and target domains must be equal.

Proper maps decompose into products of maps between individual factors.

Abstract

We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains.