Gauss curvature solitons on invariant surfaces in the homogeneous space Sol

TL;DR

This paper classifies invariant surfaces in the Sol space that serve as solitons for the Gauss curvature flow, revealing rigidity results and geometric properties of these special surfaces.

Contribution

It provides a classification of invariant Gauss curvature solitons in Sol space, including rigidity results and geometric characterizations for different invariant surfaces.

Findings

Only specific totally geodesic vertical planes are $F_1$- and $F_2$-solitons.

Main geometric properties of $F_2$- and $F_3$-solitons are established.

Rigidity results for $F_3$-invariant surfaces are proved.

Abstract

We classify invariant surfaces in the 3-dimensional solvable Lie group $\sol$ that act as solitons for the Gauss curvature flow. We consider solitons associated with the canonical basis of Killing vector fields $\{F_1, F_2, F_3\}$, where $F_1$ and $F_2$ generate horizontal translations and $F_3$ generates the scaling isometry. We establish rigidity results for $F_3$-invariant surfaces, proving that specific totally geodesic vertical planes are the only $F_1$- and $F_2$-solitons. For $F_1$-invariant surfaces, we establish the main geometric properties of $F_2$- and $F_3$-solitons in both the extrinsic and intrinsic Gauss curvature.