A Sobolev gradient flow for the area-normalised Dirichlet energy of $H^1$ maps

In this article we study the $H^1(du)$-gradient flow for the energy $E[X]=Q[X]\mathrm{/}A[X]$ where $Q[X]$ is the Dirichlet energy of $X$, $A[X]$ is the signedenclosed area of $X$, and $X:\mathbb{S}\to {\mathbb{R}}^2$ is a $H^1(du)$ map. We prove that solutions with initially positive signed enclosed area exist eternally, and converge as $t\to \mathrm{\infty }$ to a (possibly multiply-covered) circle. In this way we recover a parametrised isoperimetric inequality for $H^1(du)$ maps.