On the Shortening Rate of Collections of Plane Convex Curves by the Area-Preserving Mean Curvature Flow

Abstract

Area-preserving mean curvature flows can be used to model some phase transitions. Geometrically, in the two-dimensional case, they describe the shortening of the curves that are interfaces separating the two phases while preserving the areas of each phase, respectively. Scaling arguments suggest that under the area-preserving mean curvature flow, the rescaled total curve length $\tilde{L}(t)$ decreases as a temporal power law $\tilde{L}(t)\sim ct^{-1/2}$, where c is a positive constant. In this paper, we consider the evolution of a collection of nonintersecting smooth convex plane curves, prove a time-averaged lower bound of the decay rate of $\tilde{L}$ which exhibits the aforementioned power law, and get the dependence of the coefficient constant c on the curve shapes.