Invariant \(\lambda \)-translators in Lorentz-minkowski space

Abstract

Given \(\lambda \in \mathbb {R}\) and \({\textbf {v}}\in \mathbb {L}^3\), a \(\lambda \)-translator with velocity \({\textbf {v}}\) is an immersed surface in \(\mathbb {L}^3\) whose mean curvature satisfies \(H=\langle N,{\textbf {v}}\rangle +\lambda \), where N is a unit normal vector field. When \(\lambda =0\), we fall into the class of translating solitons of the mean curvature flow. In this paper we study \(\lambda \)-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 \(\lambda \)-translators. In the case of rotational \(\lambda \)-translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational \(\lambda \)-translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.