There exists a global conformal map between $\mathbb{R}^3$ and $S_3$: $$ g = dr^2+r^2\left(d\theta^2 + \sin^2\theta\ d\phi^2\right) $$ $$ r = R \tan \frac{\alpha}{2} $$
$$ g = \frac{R^2}{4\cos^4\frac{\alpha}{2}}\left[d\alpha^2+\sin^2\alpha\left(d\theta^2 + \sin^2\theta\ d\phi^2\right)\right] $$
There also exists a global conformal map between $\mathbb{R}^3$ and the cylinder $\mathbb{R}\times S_2$:
$$ r = R\ e^{x/R} $$
$$ g=e^{2x/R}\left[dx^2+R^2\left(d\theta^2 + \sin^2\theta\ d\phi^2\right)\right] $$
There does not appear to be a global conformal map between $\mathbb{R}^3$ and the cylinder $\mathbb{R^2}\times S_1$. At least, my attempts to find one starting from $\mathbb{R^3}$ in cylindrical coordinates: $$ g = dz^2+ d\rho^2 +\rho^2 d\phi^2 $$ and remapping $(z, \rho)$ have failed. So my questions are:
1) Is it true that there is no such map?
2) If so, how do you show that? I think that local invariants like the Cotton tensor have nothing to say, because both $\mathbb{R}^3$ and $\mathbb{R^2}\times S_1$ are flat, and instead there is some kind of global obstruction.