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

TL;DR

This paper classifies all rotational spacelike and timelike surfaces in Minkowski space that translate under the flow driven by powers of Gauss curvature, depending on the axis's causal character.

Contribution

It provides a complete classification of rotational $K^eta$-translators in Minkowski space, considering both spacelike and timelike surfaces based on the axis's causal type.

Findings

Classification of all rotational $K^eta$-translators for spacelike surfaces.

Extension of the theory to timelike rotational surfaces.

Dependence of classification on the causal character of the rotation axis.

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.