Uniqueness of Self-Similar Solutions to the Network Flow in a Given Topological Class

This paper deals with the uniqueness, within a fixed topological class, of "tree-like" solutions to the evolution of networks by curve shortening flow. More precisely, we show that if for a given initial condition, there is a solution to the network flow that is tree-like and regular for positive times, then this solution is unique within its topological class. The result in particular applies to expanding self-similar solutions. The proof is based on the following Allen-Cahn approximation result: every regular tree-like solution to the network flow can be realized as the nodal set of a family of solutions to the Allen-Cahn equation. Then, the main result of this paper follows from the uniqueness of the "ε-level" solutions. The results in this paper deal only with uniqueness of solutions. The existence of solutions for the general class of initial conditions that we consider in this paper is unknown in most cases.