TITLE: Interesting results for open Riemann surfaces
QUESTION [9 upvotes]: As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a holomorphic immersion into the complex plane. Another example is given by the theorem of Behnke and Stein, which says that every connected open Riemann surface is a Stein manifold. Is there any more interesting results for open Riemann surfaces?

REPLY [4 votes]: I hope that the following result by Bishop and myself might be "interesting": Every open Riemann surface is a cover over the sphere, branched over only three points. Equivalently, every open Riemann surface is equilaterally triangulable.
See
Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles and https://arxiv.org/abs/2103.16702 .