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 $\pi \sec \theta$ 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.

We have made some updates to the introduction. The mathematical content is unchanged from version 2