The Gradient Flow of the M\"obius energy: $\varepsilon$-regularity and consequences

TL;DR

This paper investigates the gradient flow of the M"obius energy, establishing an ps-regularity result that characterizes singularity formation and shows planar curves evolve into round circles.

Contribution

It proves an ps-regularity theorem for the M"obius energy flow and characterizes singularities, solving an open problem and demonstrating convergence of planar curves to circles.

Findings

Bounded derivatives if energy is small on a scale

Characterization of singularity formation via energy concentration

Planar curves evolve into round circles

Abstract

In this article we study the gradient flow of the M\"obius energy introduced by O'Hara in 1991. We will show a fundamental $\varepsilon$-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He. Ruling out blow-ups for planar curves, we will prove that the flow transforms every planar curve into a round circle.