I'm looking for a conformal map (not necessarily injective or surjective) from ordinary 3-dimensional euclidean space (or a region of it) to some example of nil geometry; I would be most interested if the nil space in question was compact.
So, my question is:
For some riemannian nilmanifold $N$, does a conformal map from some $A \subseteq E^3$ to $N$ always exist, or does it never exist, or is its existence dependent on other properties?