Soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space

Abstract

We show that a curve is a soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensionalMinkowski space.We show that for each fixed vector there is a 2-parameter family of soliton solutions to the flow. We prove that there are three classes of such curves. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant.