Invariant $\hat{I}»$-translators in the Heisenberg group

Abstract

We study oriented surfaces in the Heisenberg space ${\mathrm{Nil}}_3$ whose mean curvature $H$ at each point is $H=⟨N,{\partial }_z⟩+\hat{I}»$, where $N$ is the unit normal, ${\partial }_z$ is the vertical Killing vector field and $\hat{I}»\in \mathbb{R}$. These surfaces are known as $\hat{I}»$-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify $\hat{I}»$-translators invariant by the following one-parameter groups of isometries of ${\mathrm{Nil}}_3$: left-translations, rotations and helicoidal motions.