Hard analysis meets critical knots (Stationary points of the Moebius energy are smooth)
Simon Blatt, Armin Schikorra, and Philipp Reiter
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Submission date: 02. Aug. 2012
published in: Transactions of the American Mathematical Society, 368 (2016) 9, p. 6391-6438
DOI number (of the published article): 10.1090/tran/6603
with the following different title: Harmonic analysis meets critical knots : critical points of the Möbius energy are smooth
MSC-Numbers: 57M25, 46E35
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We prove that if a curve parametrized by arc length is a stationary point of the Moebius energy introduced by Jun O'Hara, then it is smooth whenever the Moebius energy is finite. Our methods, interestingly, only rely on purely analytical arguments, entirely without using Moebius invariance. Furthermore, the techniques involved are not fundamentally restricted to one-dimensional domains, but are generalizable to arbitrary dimensions.