Translating Annuli for Mean Curvature Flow

We construct a family $\mathcal{A}$ of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. For each $M$ in $\mathcal{A}$, $M$ is asymptotic as $z\to -\infty$ to four vertical planes $\{y=\pm b\}$ and $\{y=\pm B\}$ where $00$, there is an $M\in \mathcal{A}$ with inner width $b$ and with necksize $s$. (We also show that there are no translators with inner width $< \pi/2$ having the properties of the examples we construct.)