Rotational $K^{\alpha }$-translators in Minkowski space

Abstract

A spacelike surface in Minkowski space ${\mathbb{R}}_1^3$ is called a $K^{\alpha }$-translator of the flow by the powers of Gauss curvature if satisfies $K^{\alpha }=⟨N,\vec{v}⟩$, $\alpha \ne 0$, where $K$ is the Gauss curvature, $N$ is the unit normal vector field and $\vec{v}$ is a direction of ${\mathbb{R}}_1^3$. In this paper, we classify all rotational $K^{\alpha }$-translators. This classification will depend on the causal character of the rotation axis. Although the theory of the $K^{\alpha }$-flow holds for spacelike surfaces, the equation describing $K^{\alpha }$-translators is still valid for timelike surfaces. So we also investigate the timelike rotational surfaces that satisfy the same prescribing Gauss curvature equation.