conformally embedding complex tori into R^3 - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:33:07Z http://mathoverflow.net/feeds/question/20538 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20538/conformally-embedding-complex-tori-into-r3 conformally embedding complex tori into R^3 Tom Bachmann 2010-04-06T19:23:34Z 2010-04-06T19:51:44Z <p>Let $L$ be a lattice in $\mathbb{C}$ with two fundamental periods, so that $\mathbb{C}/L$ is topologically a torus. Let $p:\mathbb{C}/L \mapsto \mathbb{R}^3$ be an embedding ($C^1$, say). Call $p$ conformal if pulling back the standard metric on $\mathbb{R}^3$ along $p$ yields a metric in the equivalence class of metrics on $\mathbb{C}/L$ (i.e. a multiple of the identity matrix).</p> <blockquote> <p>Is there an explicit formula for such a p in the case of L an oblique lattice?</p> </blockquote> <hr> <h2>Backgorund</h2> <p>The existence of such $C^1$ embeddings is implied by the nash embedding theorem (fix a metric on $\mathbb{C}/L$, pick any short embedding, apply nash iteration to make it isometric and hence conformal).</p> <p>For orthogonal lattices, the solution is simple: Parametrise the standard torus of radii $r_1$, $r_2$ in the usual way. Make the ansatz $\pi(\theta, \phi) = (f(\theta), h(\phi))$, pull back the standard metric on $\mathbb{C}/L$ and solve the resulting system of ODEs. This relates $r_1/r_2$ to the ratio of the magnitudes of the periods. This shows that no standard torus can be the image of $p$ in the original question (oblique lattice), although that is geometrically clear anyway.</p> http://mathoverflow.net/questions/20538/conformally-embedding-complex-tori-into-r3/20540#20540 Answer by Sebastian for conformally embedding complex tori into R^3 Sebastian 2010-04-06T19:51:44Z 2010-04-06T19:51:44Z <p>You should have a look in Pinkall's Hopf Tori paper. You take the preimage of a curve in $S^2$ under the Hopf fibration. The lattice of the torus and hence the conformal class is then given by the generators $1\in C$ and $L+i/2 A$ (if I remember right), where $L$ is the length and $A$ is the enclosed area of the curve.</p>