Invariant $\lambda$-translators for the Gauss curvature flow in Euclidean space

TL;DR

This paper classifies special surfaces in Euclidean space that evolve under the Gauss curvature flow, focusing on those invariant under certain symmetries and satisfying a specific curvature condition involving a fixed direction and a constant.

Contribution

It provides a complete classification of $ ext{lambda}$-translators invariant under one-parameter groups of translations and rotations, extending understanding of curvature flow solutions.

Findings

Classified all invariant $ ext{lambda}$-translators under translation and rotation groups.

Identified explicit forms of surfaces satisfying the curvature condition.

Extended the theory of curvature flows with symmetry constraints.

Abstract

A $\lambda$-translator is a surface in Euclidean space $\mathbb{R}^3$ whose Gauss curvature $K$ satisfies $K=\langle N, \vec{v} \rangle +\lambda$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $\lambda \in \mathbb{R}$. In this paper, we classify all $\lambda$-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.