Smoothly embed a genus g surface in $\mathbb{R}^3$, and pick a normal vector pointing "out" of the surface at each point. Then on each tangent plane, I have a map which rotates the tangent plane 90 degrees in the direction given by the right hand rule. This gives an almost complex structure on the manifold, and every almost complex structure is integrable in real dimension 2, so this defines a complex structure on the surface.


**Question 1**: Does the above paragraph make sense? I am at the point where I think I can make grammatically correct sentences using the words above, but I am still not sure about all of the logical interconnections.


Say the first paragraph is correct. Then for each smooth embedding of the g-holed torus into $\mathbb{R}^3$ I get a complex structure on that surface. But I know that there are many different complex structures on the g-holed torus (and that the moduli space of such curves is 3g-3 complex dimensional for g>1). 


**Question 2,3,4**: How many of these complex structures can I get through different embeddings? If I give you actual equations for an embedding, can you compute which point in the moduli space I am determining? In the particular case of the torus, can you tell me which $\tau$ corresponds to a given embedding?



Does every smoothly embedded surface $\mathbb{R}^3$ inherit a natural complex structure, and if so, which one?
Question 1: Looks good to me.


Question 2: is a duplicate of [this question](https://mathoverflow.net/questions/53999/conformal-embedding-of-riemann-surfaces-into-3-space) (the answer is: every conformal structure can be so realized).


Question 3/4. There are algorithms to compute the conformal structure given a surface, due to [Sasha Bobenko](http://arxiv.org/find/nlin/1/au%3A+Bobenko_A/0/1/0/all/0/1) and his collaborators, which work by discretizing the surface, and then computing periods. Looking at the list of arxiv preprints, I am not seeing something directly on this topic, so this may be in preparation, but you can write to him.


