Does every connected, compact Riemann surface $\Sigma$ with boundary, $\partial \Sigma\not =\emptyset$, admit a holomorphic function (smooth on the boundary) $f:\Sigma\to\mathbb C$ whose derivative is everywhere non-zero?
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asked Dec 13, 2016 at 15:38
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2$\begingroup$ Even more so, every open Riemann surface admits a holomorphic immersion into ${\mathbb C}$. I will find the reference. $\endgroup$– MishaCommented Dec 13, 2016 at 15:51
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$\begingroup$ If you draw a picture of building a genus g surface with one boundary component, by attaching handles to a disk "in the standard way", you can just see an immersion to R^2. I looked a bit online for a picture of what I mean, like "pointed matched circles" in this document math.columbia.edu/~lipshitz/CambridgeSlides.pdf , an overhead picture of a highway interchange makes the point too. $\endgroup$– David TreumannCommented Dec 13, 2016 at 15:54
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1 Answer
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A more general result is proven in
Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).
As for compact surfaces with boundary, it is essentially a part of the definition that they embed holomorphically into open Riemann surfaces.
Incidentally, it is an open problem for $n$-dimensional Stein manifolds (with trivial complex tangent bundles), $n\ge 2$, if they admit locally biholomorphic immersions in ${\mathbb C}^n$. The best (to my knowledge) result in this direction is due F.Forstenič (Acta Math., 2003) who proved that every complex parallelizable $n$-dimensional Stein manifold admits a holomorphic submersion to ${\mathbb C}^{n-1}$.
