Gauss Curvature Flow Solitons on Surfaces in Walker 3-Manifolds

We initiate the study of solitons for the Gauss curvature flow on surfaces immersed in three-dimensional Lorentzian Walker manifolds. A soliton is a surface whose shape is preserved along the flow, evolving purely by the ambient isometries. Working with both the extrinsic and intrinsic Gauss curvature, we compute the fundamental forms of the relevant families of invariant surfaces and reduce the soliton condition to an ordinary differential equation on the generating curve. Our main contributions are threefold. First, we prove a rigidity theorem: a surface invariant under a one-parameter group of isometries is a soliton for that same group if and only if it is flat. Second, in the strictly Walker case we show that the only solitons with respect to the canonical parallel null field are the coordinate planes. Third, for the remaining Killing fields we classify the solitons via a phase-plane analysis of the associated ODE systems and describe the qualitative geometry of the solution surfaces. These results provide a Lorentzian analogue of the classification recently carried out by Belli and López in the Riemannian solvable Lie group Sol.