Local continuous extension of proper holomorphic maps: low-regularity and infinite-type boundaries

TL;DR

This paper establishes new conditions under which proper holomorphic maps can be continuously extended to boundaries of domains in complex space, even with low regularity and infinite-type boundary points.

Contribution

It introduces lower regularity assumptions for boundary extension and allows for infinite-type boundary points, extending previous results in complex analysis.

Findings

Proper holomorphic maps can be extended under weaker boundary regularity conditions.

Extension results apply to domains with infinite-type boundary points.

New techniques handle low-regularity boundary patches effectively.

Abstract

We prove a couple of results on local continuous extension of proper holomorphic maps $F:D \rightarrow \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $\partial{D}$ and $\partial{\Omega}$. The first result allows us to have much lower regularity, for the patches of $\partial{D}, \partial{\Omega}$ that are relevant, than in earlier results. The second result (and a result closely related to it) is in the spirit of a result by Forstneric--Rosay. However, our assumptions allow $\partial{\Omega}$ to contain boundary points of infinite type.