On the existence of translating solutions of mean curvature flow in slab regions

We prove, in all dimensions $n\ge 2$, that there exists a convex translator lying in a slab of width $\pi \sec \theta$ in ${\mathbb{R}}^{n+1}$ (and in no smaller slab) if and only if $\theta \in [0,\frac{\pi }{2}]$. We also obtain convexity and regularity results for translators which admit appropriate symmetries and study the asymptotics and reflection symmetry of translators lying in slab regions.