Simply connected translating solitons contained in slabs

In this work we show that $2$-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of $\mathbb{R}^3$ with entropy strictly less than $3$ must be mean convex and thus, thanks to a result by J. Spruck and L. Xiao, are convex. Recently, such $2$-dimensional convex translating solitons have been completely classified by Hoffman, Ilmanen, Mart\'in and White, up to an ambient isometry, as vertical plane, (tilted) grim reaper cylinders, $\Delta$-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples show that the bound on the entropy is necessary.