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

Abstract

We prove, in all dimensions $n\ge 2$, that there exists a convex translator lying in a slab of width $\pi sec𝜃$ in ${ℝ}^{n+1}$ (and in no smaller slab) if and only if $𝜃\in \left[0,\frac{\pi }{2}\right]$. 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.