Parabolic Minimal Surfaces in $\mathbb{M}^{2}\times\mathbb{R}$

Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a characterization of $\delta$-parabolicity we prove that under additional conditions on $\mathbb{M}$, an embedded minimal surface with bounded gaussian curvature is proper. The second theorem states that under some conditions on $\mathbb{M}$, if $\Sigma$ is a properly immersed minimal surface with finite topology and one end in $\mathbb{M}\times\mathbb{R}$, which is transverse to a slice $\mathbb{M}\times\{t\}$ except at a finite number of points, and such that $\Sigma\cap(\mathbb{M}\times\{t\})$ contains a finite number of components, then $\Sigma$ is parabolic. In the last result, we assume some conditions on $\mathbb{M}$ and prove that if a minimal surface in $\mathbb{M}\times\mathbb{R}$ has height controlled by a logarithmic function, then it is parabolic and has a finite number of ends.