TITLE: The cotangent bundle of a non-compact Riemann surface
QUESTION [6 upvotes]: 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 $M$.

REPLY [11 votes]: 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.