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).

Is there an explicit formula for such a p in the case of L an oblique lattice?

## Background

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).

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.

twoextra dimensions to perturb through! $\endgroup$isometricallyinto (round-metric) $\mathbb{R}\mathbb{P}^3$: via the Segre embedding $\mathbb{R}\mathbb{P}^1\times\mathbb{R}\mathbb{P}^1\to\mathbb{R}\mathbb{P}^3$. This lifts to an isometric embedding of $T^2$ into $S^3$. But $S^3\setminus \{pt\}$ is conformally equivalent to $\mathbb{R}^3$. $\endgroup$