The gradient flow of the Möbius energy near local minimizers

Abstract

In this article we show that for initial data close to local minimizers of the Mobius energy the gradient flow exists for all time and converges smoothly to a local minimizer after suitable reparametrizations. To prove this, we show that the heat flow of the Mobius energy possesses a quasilinear structure which allows us to derive new short-time existence results for this evolution equation and a Łojasiewicz-Simon gradient inequality for the Mobius energy.