# Conformal embedding of Riemann surfaces into 3-space

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 $g=1.$

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Some of the references in the MO question on Nash embedding may be relevant: mathoverflow.net/questions/37708/… . However, in that discussion, there is no mention of conformal structure. – Joseph O'Rourke Feb 1 '11 at 17:35
There has been a discussion for the tori case (mathoverflow.net/questions/20538/…). In fact, every torus can easily realised by the preimage of an appropriate curve in $S^2$ under the Hopf fibration (the construction is due to Ulrich Pinkall). – Sebastian Feb 2 '11 at 7:31
@sebastian: thanks! – Igor Rivin Feb 2 '11 at 15:23

I have thought about this question before, but at the moment I can't remember links or references. Nonetheless, many years ago I thought of a sketch of an argument that should eventually work to prove that it's always possible. Namely, start with any smooth embedding of $S$ into $\mathbb{R}^3$. This surface has some conformal structure, and any other conformal structure is described by an ellipse field well-defined up to a scale. Then it is intuitive that that you can match the ellipse field by "wrinkling" $S$, in other words by making $S$ locally wavy in the long direction of the ellipses. More rigorously, $S$ is replaced by a graph of a function that is locally sinusoidal. You don't even have to match it exactly; approximations are good enough if you can encircle the desired conformal structure in Teichmuller space. There is a similar trick for a different purpose in the video "Outside In".