Abstract:

Area-preserving curvature flows in a two-dimensional homogeneous medium have been studied for several decades. In 1986, Gage showed that an initially convex closed curve remains convex and converges to a circle as time goes to infinity. However, in many applications, the medium is not homogeneous and the closed curve moves towards a more favorable environment. The properties of the medium are described by a signal function, which is a twice continuously differentiable function defined in two-dimensional space. In this talk, we consider the area preserving flow in an inhomogeneous medium. First, I will explain the global existence of convex closed curves under certain conditions for a signal function. Next we consider the properties of stationary solutions. Finally I will explain the dynamics of curves when the areas are small.