Rigidity of solitons of the Gauss curvature flow in Euclidean space

TL;DR

This paper classifies special solutions called solitons of the Gauss curvature flow in Euclidean space, showing that only certain geometric shapes like planes, spheres, and cylinders have constant mean curvature.

Contribution

It provides a complete classification of $mbda$-translating solitons and $mbda$-shrinkers with constant mean curvature, including those with one constant principal curvature.

Findings

Planes and cylinders are the only $mbda$-translating solitons with constant mean curvature.

Planes, spheres, and cylinders are the only $mbda$-shrinkers with constant mean curvature.

Classification of solitons with one constant principal curvature.

Abstract

In this paper, we consider $\lambda$-translating solitons and $\lambda$-shrinkers of the Gauss curvature flow in Euclidean space. We prove that planes and circular cylinders are the only $\lambda$-translating solitons with constant mean curvature. We also prove that planes, spheres and circular cylinders are the only $\lambda$-shrinkers with constant mean curvature. We give a classification of the $\lambda$-translating solitons and $\lambda$-shrinkers with one constant principal curvature.