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