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
1 Answer
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 926 on page 22 .

$\begingroup$ Thank you very much. In case $Y$ has dimension 1, it is a connected Riemann Surface, but this doesn't imply that is has connected boundary. Can't we say anything in this case? Just about $Y$, no matter of subsets. Thanks $\endgroup$– JoeOct 4, 2021 at 21:23

1$\begingroup$ You are welcome. Could you clarify your question . When Y is a Riemann surface it can have uncountably infinite number of ends . $\endgroup$ Oct 4, 2021 at 23:15