An isoperimetric flow in the plane
Abstract: We consider a fourth order flow of compact curves in the plane. This is the gradient flow of arclength on the space of curves bounding a fixed area, with a particular metric (not the classic L2 metric) on the space of such curves. While this was first studied as a toy problem meant to build momentum for studying a gradient flow on compact lagrangian submanifolds in higher dimensions, it takes some work to get any result for the one dimensional flow. The approach is reminiscent of the work of Gage and Hamilton on curve shortening flow, without the help of a maximum principle for fourth order equations.
If the curve is near enough to the circle in a C1 sense, the curve converges back to a circle.