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.