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 $K^{\alpha} = \langle N, \vec{v} \rangle$, $\alpha \neq 0$, 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.