Let
${\mathbb{M}}^{2}$
be a complete noncompact orientable surface of nonnegative curvature.
We prove 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
$\mathbb{M}\times \left\{t\right\}$
except at a finite number of points, and such that
$\Sigma \cap \left(\mathbb{M}\times \left\{t\right\}\right)$ 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.
