§ 3.29 Invariant translators in the Heisenberg group750539FAA8AF

For the Heisenberg metric $g_\lambda =dx^2+dy^2+\lambda (dz+(y\, dx-x\, dy)/2)^2$, $\lambda >0$, Pipoli constructs and classifies invariant vertical translators with unit Killing field $\lambda ^{-1/2}\partial _ z$. The families include nonconvex complete slab graphs, a rotational entire bowl, embedded rotational catenoids with positive neck radius, and helicoidal examples. The catenoids have two ends; the bowl has one. The helicoidal generating curve has two individually embedded spiraling arms, whose union can have infinitely many intersections, giving nonembedded surfaces. Thus embeddedness must be recorded separately for each branch. The rotational asymptotics have leading height $r^2/(2\sqrt\lambda )$, with logarithmic or power corrections depending on whether $\lambda <4$, $\lambda =4$ or $\lambda >4$. This leading term is not an exact formula for the bowl. The flow is generated by vertical left translations for all real time. A Euclidean drawing of these coordinate graphs does not encode the ambient metric or its second fundamental form. The figure labels the metric parameter and compares the invariant orbit models at three vertical-translation times; it is schematic, including the unresolved nonlinear profiles.

Sources: [Pipoli2018].