Invariant $\hat{I}»$-translators in ${\mathbb{S}}^2\times \mathbb{R}$

Abstract

A $\hat{I}»$-translator in ${\mathbb{S}}^2\times \mathbb{R}$ is an oriented surface whose mean curvature $H$ satisfies $H=⟨N,{\partial }_z⟩+\hat{I}»$, where $N$ is the unit normal, ${\partial }_z$ is the vertical Killing vector field tangent to the fibers of the submersion and $\hat{I}»\in \mathbb{R}$. When $\hat{I}»=0$ we fall into the class of translators. In this paper, we classify all $\hat{I}»$-translators that are invariant by a one-parameter group of rotations and by vertical translations of ${\mathbb{S}}^2\times \mathbb{R}$.