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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

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    $\begingroup$ You need to assume Y has complex dimension at least two. $\endgroup$ Oct 4, 2021 at 15:59

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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 .

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  • $\begingroup$ 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 $\endgroup$
    – Joe
    Oct 4, 2021 at 21:23
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    $\begingroup$ 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

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