Translating Annuli for Mean Curvature Flow

We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\mathrm{\infty }$ to four vertical planes $\{y=\pm b\}$ and $\{y=\pm B\}$, where $0<b\le B<\mathrm{\infty }$. We call $b$ and $B$ the inner width and the (outer) width of the translator. We show that for each $b\ge \pi \mathrm{/}2$ and each $s>0$, there is a translator in the family with inner width $b$ and with necksize $s$. (We also show that there are no translators with inner width $<\pi \mathrm{/}2$ having the properties of the examples we construct.)