§ 3.5 Translators from grim-reaper configurations (X. H. Nguyen, 2010; X. H. Nguyen, 2012)E83F92A45A1F

Take finitely many translated grim-reaper cylinders $y=-\log \cos (x-b_ j)+c_ j$ in $\mathbb R^3$, with no coincident asymptotic planes and no line common to three cylinders. Assign a positive integer $m_ k$ to each intersection line. For all sufficiently small $\tau >0$, Nguyen constructs a complete embedded translator with velocity $e_ y$, reflection symmetry in $z=0$, and period $2\pi \tau $ in $z$. The smallness bound depends on the configuration, its separation and intersection angles, and the integers $m_ k$. Away from intersection lines it converges to the grim cylinders as $\tau \downarrow 0$; at each line the rescaled limit is a singly periodic Scherk surface with the prescribed number of periods. The periodic extension starts with finitely many cylinders per period vector $a=(a_ x,a_ y,0)$, requires uniformly separated asymptotic planes and no triple intersection, and gives two independent periods. In particular suitable configurations exist for $a_ x\ne 0$ and $a_ x\ne \pm \pi /q$, $q\in \mathbb N$. The flow is $\Sigma +te_ y$, $t\in \mathbb R$. Genus and ends of the full periodic surface must be distinguished from those of a quotient. The drawing shows the exact limiting grim pieces and marked gluing locations; it does not numerically resolve the Scherk bridges or the elliptic correction.

Sources: [Nguyen2013, Nguyen2015].