Invariant $\hat{I}»$-translators in Lorentz-Minkowski space

Abstract

Given $\hat{I}»\in \mathbb{R}$ and $\text{v}\in {\mathbb{L}}^3$, a $\hat{I}»$-translator with velocity $\text{v}$ is an immersed surface in ${\mathbb{L}}^3$ whose mean curvature satisfies $H=⟨N,\text{v}⟩+\hat{I}»$, where $N$ is a unit normal vector field. When $\hat{I}»=0$, we fall into the class of translating solitons of the mean curvature flow. In this paper we study $\hat{I}»$-translators in ${\mathbb{L}}^3$ that are invariant under a 1-parameter group of translations and rotations. The former are cylindrical surfaces and explicit parametrizations are found, distinguishing on the causality of both the ruling direction and the $\hat{I}»$-translators. In the case of rotational $\hat{I}»$-translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational $\hat{I}»$-translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.