The gradient flow of the Möbius energy: $\varepsilon$-regularity and consequences

Abstract

We study the gradient flow of the Möbius energy introduced by O’Hara (Topology 30:2 (1991), 241–247). We will show a fundamental $𝜀$-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 (