On the connectedness of the boundary of $q$-complete domains

TL;DR

This paper proves that the boundary of any relatively compact $q$-complete domain in complex and almost complex manifolds of dimension at least two is connected, without requiring boundary regularity assumptions.

Contribution

It extends the connectedness result from Stein domains to $q$-complete domains and to almost complex manifolds, broadening the scope of boundary connectedness theorems.

Findings

Boundaries of $q$-complete domains are connected in complex manifolds.

The connectedness result applies without boundary regularity assumptions.

The proofs are valid in the context of almost complex manifolds.

Abstract

The boundary of every relatively compact Stein domain in a complex manifold of dimension at least two is connected. No assumptions on the boundary regularity are necessary. The same proofs hold also for $q$-complete domains, and in the context of almost complex manifolds as well.