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The gradient flow of
the Möbius energy: $\varepsilon$-regularity and
consequences
Simon Blatt |
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Vol. 13 (2020), No. 3, 901–941
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Abstract |
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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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Keywords
Möbius energy, geometric evolution equations, gradient
flow, long-time existence
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Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 35S10
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Milestones
Received: 30 October 2018
Accepted: 7 March 2019
Published: 15 April 2020
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Authors |
| Simon Blatt | |
| Paris Lodron Universität
Salzburg Salzburg Austria |
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