  We prove that any complete immersed two-sided mean convex translating soliton
$\Sigma \subset \mathbb{R}^3$ for the mean curvature flow is convex. As a
corollary it follows that an entire mean convex graphical translating soliton
in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the
mean curvature of $\Sigma$ tends to zero at infinity, then $\Sigma$ can be
represented as an entire graph and so is the "bowl soliton". Finally we
classify all locally strictly convex graphical translating solitons defined
over strip regions.

Differential Geometry