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Suppose that $M$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $M$ is trivial. Therefore there exists a holomorphic 1-form $\omega$ that does not have any zeroes on $M$. I am interested to know whether we can find an exact 1-form $\omega=df$, where $f$ is some holomorphic function on $M$, such that $\omega$ does not have zeroes on $M$.

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Such $f$ exists on every open Riemann surface:

R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces. Math. Ann., 174:103–108, 1967.

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    $\begingroup$ We discussed this paper at mathoverflow.net/questions/287275 $\endgroup$
    – David E Speyer
    Commented Dec 5, 2018 at 19:30
  • $\begingroup$ @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 6, 2018 at 0:13

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