Classification of invariant Gauss curvature solitons in the Heisenberg space $\nil$

Abstract

In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector fields and the three families of invariant surfaces (vertical translations, horizontal translations, and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.