For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $u$. Moreover, by the Gauss-Bonnet theorem the sign of the curvature of $\overline{g}$ is uniquely determined by the topology of $S$.

However, when $S$ is non-compact, it may admit different kinds of complete metrics of constant curvature. For example, on $\mathbb{R}^2$ you have many flat metrics but you also have the hyperbolic metric coming from some identification of the plane with the unit disk. On the cylinder $C = \mathbb{R}^2 \backslash \mathbb{Z}$, there is the natural flat metric coming from the Euclidean metric but there is also the hyperbolic metric arising from the quotient of the upper half-plane by a horizontal translation.

I am interested in formulating some (geometric) criterion for a metric $g$ on $\mathbb{R}^2$ or on the cylinder $C$ to be conformal to a flat metric. For example, if $(C,g)$ is asymptotic to the product metric $dx^2 + d\theta^2$ with suitable decay assumptions, I was shown how to solve the PDE for a conformal deformation to a flat metric using weighted Sobolev spaces. But for weaker notions of "asymptotic flatness" of $g$, this method wouldn't work although I feel like such a metric should be conformally flat, even for very weak notions of asymptotic flatness...

So, given a metric $g$ on the plane or on the cylinder, how can I differentiate between $g$ being conformally flat or conformally hyperbolic?