Soliton solutions to the inverse curvature flow on the de Sitter surface $d{S}_2$

In geometric flows, soliton solutions evolve by isometries of the ambient surface. In this paper, we study soliton solutions to the inverse curvature flow of regular curves on the de Sitter surface $d{S}_2$, a maximally symmetric Lorentzian manifold of constant positive curvature embedded in ${\mathbb{R}}_1^3$. We show that a regular curve is a soliton solution if and only if its inverse geodesic curvature equals the Lorentzian inner product of its tangent vector field with a fixed nonzero vector $\mathbf{v}\in {\mathbb{R}}_1^3$. On the basis of this characterization, we prove the existence of a two-parameter family of nontrivial soliton solutions for each $\mathbf{v}$, including both spacelike and timelike curves. Moreover, we completely classify both spacelike and timelike soliton solutions to the inverse curvature flow on the de Sitter surface. The classification is determined by the causal character of $\mathbf{v}$. For spacelike solitons, each solution is either complete or has a finite singular endpoint at which the geodesic curvature blows up, while the other end escapes to infinity. Moreover, complete spacelike solitons exist only when $\mathbf{v}$ is spacelike. By contrast, timelike solitons may be complete for any causal character of $\mathbf{v}$. In the incomplete case, each solution possesses a finite endpoint whose geometry depends on the causal character of $\mathbf{v}$. In all cases, we determine the asymptotic behavior of the curves and their curvatures at every nonsingular end. We also present several explicit examples and visualizations illustrating the different types of soliton solutions.