Consider the usual cylinder $S^1 \times [0, 1]$ embedded in $\mathbb{R}^3$. I am interested in knowing what are the surfaces in $\mathbb{R}^3$ that are conformal to this cylinder. If this were a closed surface, I would know how to attack the problem. One of my aims is to get a better understanding of the "Uniformization theorem" in different settings.
Edit: By two diffeomorphic surfaces $(M_1, g_1)$ and $(M_2, g_2)$ ($g_i$ are metrics) being conformal, I mean $g_2 = e^\phi g_1$. As far as I can see, this does not necessarily have to extend to a conformal transformation of the ambient space.