Self similar expanding solutions of the planar network flow

We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schn\"urer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect to a complete negatively curved metric on the unit ball which span $k$ specified points on the boundary at infinity. We also provide a sharp formulation of the regularity of these solutions at $t=0$.