Abstract

In this article we consider the length functional defined on the space of
immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve
shortening flow as the gradient flow of the length functional. Motivated by the
triviality of the metric topology in this space, we consider the gradient flow
of the length functional with respect to the $H^1(ds)$metric. Circles with
radius $r_0$ shrink with $r(t) = \sqrt{W(e^{c2t})}$ under the flow, where $W$
is the Lambert $W$ function and $c = r_0^2 + \log r_0^2$. We conduct a thorough
study of this flow, giving existence of eternal solutions and convergence for
general initial data, preservation of regularity in various spaces, qualitative
properties of the flow after an appropriate rescaling, and numerical
simulations.