Rigidity of solitons of the Gauss curvature flow in Euclidean space

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