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

TL;DR

This paper proves the existence of convex translating solutions to mean curvature flow within specific slab regions in all dimensions, characterizing when such solutions exist based on the slab width and symmetry properties.

Contribution

It establishes the precise conditions for the existence of convex translators in slabs, including symmetry, convexity, regularity, and asymptotic behavior, in all dimensions.

Findings

Convex translators exist in slabs of width π sec θ if and only if θ ∈ [0, π/2].

Convexity and regularity results are obtained for symmetric translators.

Asymptotic and reflection symmetry properties of translators are studied.

Abstract

We prove, in all dimensions $n\geq 2$, 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.