This paper is a contribution to the problem of embedding Riemann surfaces properly into \(\mathbb{C}^{2}\). It is known that for \(d \geq 2\), the number \([3d/2]+1\) gives the smallest dimension \(N_{d}\) so that every \(d\)-dimensional Stein manifold can be embedded properly into \(\mathbb{C}^{N_{d}}\). The case \(d=1\), i.e., the question of embedding a one-dimensional Stein manifold properly into \(\mathbb{C}^{2}\), is open.
In the paper under review, the author proves the following result. Let \(\mathbb{T}\) be a one-dimensional complex torus and consider subsets with finitely many boundary components, none of which are points. Such subsets can be properly embedded into \(\mathbb{C}^{2}\).
The author also shows that the algebras of analytic functions on certain countably connected subsets of closed Riemann surfaces are doubly generated.