Non-Existence of Theta-Shaped Self-Similarly Shrinking Networks Moving by Curvature

We prove that there are no networks homeomorphic to the Greek "theta" letter (a double cell) embedded in the plane with two triple junctions with angles of $120$ degrees, such that under the motion by curvature they are self-similarly shrinking. This fact completes the classification of the self-similarly shrinking networks in the plane with at most two triple junctions.