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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asked Feb 1, 2011 at 17:21
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$\begingroup$ 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. $\endgroup$– Joseph O'RourkeCommented Feb 1, 2011 at 17:35
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4$\begingroup$ 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). $\endgroup$– SebastianCommented Feb 2, 2011 at 7:31
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2 Answers
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The embedding question has been answered in the affirmative by Adriano Garsia in 1961. Here is the link to MR. In fact, he has shortly after shown that the embedding might be chosen with real algebraic image.
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3$\begingroup$ Thanks! I have discussed this question with many experts, and everyone thought the question was open. A lesson to those of us who think people will remember our work (according to MR, A. Garsia seems very much alive and well, still). $\endgroup$ Commented Feb 1, 2011 at 18:13
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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".
Of course it's messy to make all of the required transitions between the patches of wrinkles on the surface. Moreover the surface is not flat, only flat to first order, and a function cannot be graphed rectilinearly, only approximately so. I think that Garsia's paper uses the same idea or a similar idea, but he slogs through all of the approximations to make it work.
answered Feb 1, 2011 at 17:53