Translators of the Gauss curvature flow

TL;DR

This paper classifies all rotationally symmetric and helicoidally invariant translators under the Gauss curvature flow, providing explicit descriptions of these surfaces in Euclidean space.

Contribution

It provides a complete classification of $K^eta$-translators with symmetry, including rotational and helicoidal cases, under the Gauss curvature flow.

Findings

Existence of a $K^eta$-translator intersecting the rotation axis for each $eta$.

Explicit descriptions of helicoidal $K^eta$-translators.

Classification of translators obtained via separation of variables.

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.