For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:

• Riemann Sphere $\mathbb{C}\cup\{\infty\}$ (elliptic type).
• Complex plane (parabolic type).
• Open unit disk (hyperbolic type).

On the other hand, given a Riemann surface one can asks if it is hyperbolic in the Gromov's sense. In other words, does there exists $\delta>0$ such that all the geodesic triangles in the surface are $\delta$-thin?

It seems to me that this two notions of hyperbolicity are not equivalent and one can have counterexamples in both directions. For instance, the two dimensional torus $\mathbb{T}^2$ is hyperbolic in Gromov's sense (since it is compact) but it is conformally equivalent to the complex plane and therefore of parabolic type.

My questions are: what is a necessary condition to guarantee that a surface of hyperbolic type is also Gromov's hyperbolic? what is known about the relation of these two notions?

Related Question: Let $G$ be an infinite planar graph with uniformly bounded degree and assume that the simple random walk is transient. Is the graph necessarily Gromov's hyperbolic?