Let $Y$ be a Stein manifold and $D\subset\subset Y$ be a Stein domain. I think $\overline D$ has connected boundary, and it should be somewhere, but I cannot find a reference for this. Thanks



Connectedness of boundary of a Stein domain
This is a consequence of the fact that any Stein manifold with complex dimension at least 2 has one end. This follows from Hartogs extension across compact sets in Stein manifolds. You can find a proof of this Hartogs
extension in the paper of Serre on Serre duality Commentarii Mathematici Helvetici volume 29 1955 pages 9-26 on page 22 .


