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

  • 2
    $\begingroup$ You need to assume Y has complex dimension at least two. $\endgroup$ Oct 4, 2021 at 15:59

1 Answer 1


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 .

  • $\begingroup$ 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
  • 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.