Paper · May 16, 2026

Gauss curvature solitons on invariant surfaces in the homogeneous space Sol

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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.

Comments: 28 pages, 5 figures

Other curvature flows Gauss curvature flow 3-manifolds Homogeneous spaces Stability & uniqueness

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Gauss curvature solitons on invariant surfaces in the homogeneous space Sol

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