On the $H^1(ds)$-gradient flow for the length functional

TL;DR

This paper investigates the gradient flow of the length functional on immersed planar curves using an $H^1(ds)$-metric, revealing unique shrinking behaviors, existence of eternal solutions, and convergence properties.

Contribution

It introduces the $H^1(ds)$-gradient flow for the length functional, providing new insights into curve evolution beyond the classical $L^2(ds)$-based flow.

Findings

Circles shrink according to a Lambert W function formula.

Existence of eternal solutions and convergence for general initial data.

Preservation of regularity and qualitative flow properties.

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 $r(t) = \sqrt{W(e^{c-2t})}$ 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.