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We show that a curve is a soliton solution to the curve shortening flow ifand only if its geodesic curvature can be written as the inner product betweenits tangent vector field and a fixed vector of the 3-dimensional Minkowskispace. We use this characterization to provide a qualitative study of thesolitons. We show that for each fixed vector there is a 2-parameter family ofsoliton solutions to the curve shortening flow on the 2-dimensional hyperbolicspace. Moreover, we prove that each soliton is defined on the entire real line,it is embedded and its geodesic curvature converges to a constant at each end.

Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane