Vol. 13, No. 3, 2020

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

Simon Blatt

Vol. 13 (2020), No. 3, 901–941
DOI: 10.2140/apde.2020.13.901
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 (Comm. Pure Appl. Math. 53:4 (2000), 399–431).

Ruling out blow-ups for planar curves, we will prove that the flow transforms every planar curve into a round circle.

Keywords
Möbius energy, geometric evolution equations, gradient flow, long-time existence
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 35S10
Milestones
Received: 30 October 2018
Accepted: 7 March 2019
Published: 15 April 2020
Authors
Simon Blatt
Paris Lodron Universität Salzburg
Salzburg
Austria