Does every non-compact Riemann surface embed holomorphically into $\mathbb{C}^2$?
Question: Can every non-compact Riemann surface be holomorphically embedded into $\mathbb{C}^2$? If not, what are some (all?) of the obstructions to such an embedding?
This question is partially inspired by the Wikipedia page on Stein manifolds, which taught me two things:
Behnke-Stein Theorem (1948): Every non-compact Riemann surface is Stein, hence can be holomorphically embedded in some $\mathbb{C}^N$.
Every Stein manifold of complex dimension $n$ can be embedded into $\mathbb{C}^{2n+1}$ by a biholomorphic proper map. (It would be nice to have a citation for this.)
Together, these two theorems imply that every non-compact Riemann surface holomorphically embeds into $ \mathbb{C}^3$. This raises the question of embedding into $\mathbb{C}^2$.
This is an important open problem going back to Forster, Bell and Narasimhan. To quote from
A. Alarcón, F. Forstnerič, Every bordered Riemann surface is a complete proper curve in a ball, Mathematische Annalen, Vol. 357 (2013) Issue 3, 1049–1070.
"It is classical that any open Riemann surface immerses properly holomorphically into ${\mathbb C}^2$, but it is an open question whether it embeds into ${\mathbb C}^2$."
Existence of proper and non-proper embeddings are both unknown. Many things are known. For instance, the complement to any finite collection of pairwise disjoint closed topological disks in a genus 1 compact Riemann surface (aka a torus or an elliptic curve) embeds:
E. F. Wold, Embedding subsets of tori properly into ${\mathbb C}^2$. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1537–1555.
It is also known that there are no topological obstructions to a holomorphic embedding of an open Riemann surface in ${\mathbb C}^2$:
A. Alarcón, F. López, Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into ${\mathbb C}^2$. J. Geom. Anal. 23 (2013), no. 4, 1794–1805.
and
A. Alarcón, J. Globevnik, Complete embedded complex curves in the ball of ${\mathbb C}^2$ can have any topology. Anal. PDE 10 (2017), no. 8, 1987–1999.
See these slides for further background on this problem.