Self-Similarity in Two-Phase Curvature Flow

In this work, we present a family of exact self-similar solutions for the distribution of enclosed areas under mean curvature flow on the union of disjoint closed curves in the plane. To our knowledge, this is the first example of closed-form exact self-similar solutions for the mean curvature flow. We perform numerical simulations for large sets of initial data obtained as the zero level set of Gaussian random fields and observe numerical convergence to one particular member of this family.