Dynamics of area-preserving curvature flow of convex plane curves with small area in an inhomogeneous medium

Abstract

In 1986, Gage studied area-preserving curvature flows in a two-dimensional homogeneous medium and proved that an initially convex closed curve remains convex and converges to a circle as time approaches infinity. However, a medium may not be homogeneous in many applications such as cell motility and motions of active matters. Therefore, this work introduces an area-preserving curvature flow in an inhomogeneous medium, which is a natural extension of an inhomogeneous medium. Through this extension, a closed curve numerically moves towards the higher medium by the flow. To show this fact, the case is considered in which the area enclosed by the closed curve is small. In this situation, it is shown that the dynamics of its center is approximated by a gradient flow determined by the distribution of the inhomogeneous medium and that its center moves toward the critical point of the inhomogeneous medium.