Translators of the Gauss curvature flow

Abstract

A $K^{\alpha }$-translator is a surface in Euclidean space $\r^3$ that moves by translations in a spatial direction and under the $K^{\alpha }$-flow, where $K$ is the Gauss curvature and $\alpha$ is a constant. We classify all $K^{\alpha }$-translators that are rotationally symmetric. In particular, we prove that for each $\alpha$ there is a $K^{\alpha }$-translator intersecting orthogonally the rotation axis. We also describe all $K^{\alpha }$-translators invariant by a uniparametric group of helicoidal motions and the translators obtained by separation of variables.