Embedding open connected Riemann surfaces in $\mathbb{C}^2$

Question

This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is being asked here since it appears to be at a "research level".

Is there a counter-example (or an obstruction) to embedding an open connected Riemann surface into $\mathbb{C}^2$?

A theorem of Narasimhan (Narasimhan, R. "Imbedding of open Riemann surfaces", Göttingen Nachrichten, No. 7 (1960), pp. 159-165; also see American Journal of Mathematics Vol. 82, No. 4 (Oct., 1960), pp. 917-934) proves that any open connected Riemann surface has a non-singular embedding in $\mathbb{C}^3$.

Extending this to $\mathbb{C}^2$ seems to be difficult as a linear projection would introduce nodal singularities in general.

There are examples of affine algebraic curves that do not embed algebraically into $\mathbb{A}^2$. One obstruction in this case is that for a smooth curve $X$ in $\mathbb{A}^2$, we have $\Omega^1_{X}$ is a trivial line bundle, whereas there are affine algebraic curves with non-trivial $\Omega^1_{X}$. However, this is not an obstruction in the case of an open Riemann surface since line bundles on it are holomorphically trivial!

Answer 1

This is an open problem, known as

Bell-Narasimhan conjecture. Every open Riemann surface admits a proper holomorphic embedding into $\mathbb{C}^2$.

Look at F. Forstnerič's book Stein Manifolds and Holomorphic Mappings, Problem 9.10.1 p. 446.

Answer 2

One of the first example of proper embedding of open Riemann surface is,

Proper embedding of open unit disk in $\mathbb C^2$ which is one of corollary of Fatou-Bieberbach domains (holomorphic dynamics in $\mathbb C^2$). There are results about proper embedding of annulus as well.

But general question is still open which is known as Bell-Narasimman conjecture.

Bell-Narasimman conjecture has two parts (both are open) -

1) Every open Riemann surface can be embedded in $\mathbb C^2$

2) Every embedded open Riemann surface can be properly embedded in $\mathbb C^2$.