Motivation and examples of parabolic manifolds

Question

let $(M^{n},g)$ a Riemannian manifold, we say that $M$ is parabolic if constant functions over M are the only subharmonic functions which are upper bound, i.e, for a function $u \in C^{2}(M)$, we have $\Delta u \geq 0$ and $u\leq u^{*}<\infty$ then $u$ is constant. Liouville theorem for subharmonic functions asserts that $\mathbb{R}^{2}$ is a parabolic manifold.

I would like two things about this definition:

1. What is motivation for study of parabolic manifolds?

   2. If $N$ is a complete Riemannian manifold , then $\mathbb{R}^{2} \times N$ is parabolic? Why?