Complete translating solitons to the mean curvature flow in ℝ3 with nonnegative mean curvature

Abstract

We prove that any complete immersed two-sided mean convex translating soliton $\Sigma\subset{\Bbb R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in ${\Bbb R}^3$ 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 the asymptotic behavior of all locally strictly convex graphical translating solitons defined over strip regions.