I can't seem to find any work on the following question: Can every (closed, of finite type) Riemann surface $S$ be realized as an embedded (or even immersed) smooth surface in Euclidean $3$-space, where by realized, I mean that the induced conformal structure on $T \subset \mathbb{E}^3$ is the conformal structure on $S?$ The answer is obviously yes when $S$ is of genus $0,$ but that's where the obvious statements end -- I don't know the answer for $g=1.$ 



Conformal embedding of Riemann surfaces into 3-space
The embedding question has been answered in the affirmative by Adriano Garsia in 1961. Here is the [link](http://www.ams.org/mathscinet-getitem?mr=125591) to MR. In fact, he has [shortly after shown](http://www.ams.org/mathscinet-getitem?mr=148898) that the embedding might be chosen with real algebraic image.


