On a Third Order Flow of Convex Closed Plane Curves

Abstract

We study a curve flow for convex closed plane curves. It is described by a third order linear equation for the radius of curvature of the evolving curve. It is shown that under the flow the evolving curve stays convex, bounds fixed area, length, and has fixed center. However, its curvature may blow up in finite time.

If the curvature of this flow does not blow up before time $2\pi$, then the flow will exist smoothly for all time and is periodic in time with period $2\pi$. In particular, the flow does not have a limiting curve unless the initial curve is a circle.